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Elements of Vasiliev theory

V.E. Didenko∗ and E.D. Skvortsov∗†

∗Lebedev Institute of Physics, Moscow, Russia

† Albert Einstein Institute, Potsdam, Germany

Abstract

We propose a self-contained description of Vasiliev higher-spin theories withthe emphasis on nonlinear equations. The main sections are supplemented withsome additional material, including introduction to gravity as a gauge theory; thereview of the Fronsdal formulation of free higher-spin fields; Young diagrams andtensors as well as sections with advanced topics. The general discussion is dimensionindependent, while the essence of the Vasiliev formulation is discussed on the baseof the four-dimensional higher-spin theory. Three-dimensional and d-dimensionalhigher-spin theories follow the same logic.

∗didenko@lpi.ru†skvortsov@lpi.ru

1

Contents

1 Introduction 4

2 Metric-like formulation for free HS fields 72.1 Massless fields on Minkowski background . . . . . . . . . . . . . . . . . . . 72.2 Massless fields on (anti)-de Sitter background . . . . . . . . . . . . . . . . 12

3 Gravity as gauge theory 143.1 Tetrad, Vielbein, Frame, Vierbein,.... . . . . . . . . . . . . . . . . . . . . . 153.2 Gravity as a gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Unfolding gravity 31

5 Unfolding, spin by spin 385.1 s = 2 retrospectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 s ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 s = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 Zero-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.6 All spins together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Unfolding 556.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Higher-spin theory in four dimensions 63

8 Vector-spinor dictionary 638.1 so(3, 1) ∼ sl(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648.2 Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

9 Free HS fields in AdS4 679.1 Massless scalar and HS on Minkowski . . . . . . . . . . . . . . . . . . . . . 689.2 Spinor version of AdS4 background . . . . . . . . . . . . . . . . . . . . . . 709.3 Extension to AdS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729.4 HS gauge potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

10 Higher-spin algebras 76

11 Vasiliev equations 8711.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8711.2 Quasi derivation of Vasiliev equations . . . . . . . . . . . . . . . . . . . . . 8911.3 Extended star-product, Klein operators, HS equations . . . . . . . . . . . . 9111.4 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.5 Manifest Lorentz symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.6 Higher orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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12 Extras 10812.1 Fronsdal operator on Riemannian manifolds . . . . . . . . . . . . . . . . . 10812.2 MacDowell-Mansouri-Stelle-West . . . . . . . . . . . . . . . . . . . . . . . 10912.3 Interplay between diffeomorphisms and gauge symmetries . . . . . . . . . . 11312.4 Chevalley-Eilenberg cohomology and interactions . . . . . . . . . . . . . . 11612.5 Universal enveloping realization of HS algebra . . . . . . . . . . . . . . . . 11812.6 Advanced ⋆-products: Cayley transform . . . . . . . . . . . . . . . . . . . 12012.7 Poincare Lemma, Homotopy integrals . . . . . . . . . . . . . . . . . . . . . 121

A Indices 123

B Multi-indices and symmetrization 123

C Solving for spin-connection 124

D Differential forms 124

E Young diagrams and tensors 126E.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126E.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132E.3 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

F Symplectic differential calculus 136

G More on so(3, 2) 137G.1 Restriction of so(3, 2) to so(3, 1) . . . . . . . . . . . . . . . . . . . . . . . . 137G.2 so(3, 2) ∼ sp(4,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3

1 Introduction

One of the goals of quantum field theory is to explore the landscape of consistent theories.Normally one starts with some set of free, noninteracting fields and then tries to makethem interact. Given the fact that the free fields are characterized by spin and mass wecan ask the following question: which sets of fields specified by their spins and massesadmit consistent interaction? Massless fields of spins s = 1, 3

2, 2, ... being gauge fields

are of particular interest because their interactions are severely constrained by gaugesymmetry. The well known examples include: Yang-Mills theory as a theory of masslessspin-one fields; the gravity as a theory of a massless spin-two; supergravities as theories ofa number of spin-3

2fields, graviton and possibly some other fields required for consistency;

string theory which spectrum contains infinitely many fields of all spins, mostly massivebeing highly degenerate by spin.

There is a threshold value of spin, which is 2. Once a theory contains fields of spinsnot greater than two its spectrum can be finite. If there is at least one field of spin greaterthan two, a higher-spin field, the spectrum is necessarily infinite, containing fields of allspins. String theory is an example of such theory. The Vasiliev higher-spin theory isthe missing link in the evolution from the field theories of lower spins, s ≤ 2, to stringtheories. The Vasiliev theory is the minimal theory whose spectrum contains higher-spinfields. Its spectrum consists of massless fields of all spins s = 0, 1, 2, 3, ..., each appearingonce in the minimal theory, and in this respect it is much simpler than string theory. Theimportant difference of Vasiliev theory in comparison with string theory is that while thelatter has dimensionful parameter α′, the former does not being the theory of gauge fieldsbased on the maximal space-time symmetry. This higher-spin symmetry is an ultimatesymmetry in a sense that it can not result from spontaneous symmetry breaking. TheVasiliev theory therefore has no energy scale and can be thought of as a toy model of thefundamental theory beyond Planck scale.

In this note we would like to give a self-contained review on some aspects of higher-spin theory. The subject dates back to the work of Fronsdal, [1], who has first foundthe equations of motion and the action principle for free massless fields of arbitrary spin.His equations naturally generalize those of Maxwell for s = 1 and the linearized Einsteinequations for s = 2. At the same time, as spin gets larger than three (in bosonic case) theFronsdal fields reveal certain new features. It is a subject of many no-go theorems statingthat interacting higher-spin fields cannot propagate in Minkowski space. Once a yes-goresult is available thanks to Fradkin and Vasiliev, [2–4] we are not going to discus thesetheorems in any detail referring to excellent review [5]. The crucial idea that allowed oneto overcome all no-go theorems was to replace flat Minkowski background space with theanti-de Sitter one, i.e. to turn on the cosmological constant. That higher-spin theoryseems to be ill-defined on the Minkowski background is not surprising from the point ofview of absence of the dimensionful parameter. In other words the interaction verticesthat carry space-time derivatives would be of different dimension in that case. The AdSbackground simply introduces such a dimensionful parameter, the cosmological constant.

A canonical way of constructing interacting theories is order by order. In the realmof perturbative interaction scheme one begins with a sum of quadratic Lagrangians for agiven spectrum of spins and masses (these are zero since we consider gauge fields) and

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tries to deform them by some cubic terms while maintaining the gauge invariance, thenby some quartic terms, etc. The gauge transformations perturbatively get deformed aswell. If a nontrivial solution to cubic deformations is found we are said to have cubicinteraction vertices. A lot is known about cubic interactions both in the metric-likeapproach of Fronsdal [6–23] and in the frame-like approach of Vasiliev and Fradkin-Vasiliev[2–4, 24–30].

The cubic level, however, is insensitive to the spectrum of fields, i.e. given a set ofconsistent cubic couplings among various fields one can simply sum up all cubic deforma-tions into a single Lagrangian, which is again consistent up to the cubic level. For thesimplest case of a number of spin-one fields one finds that the cubic consistency relies onsome structure constants fabc being antisymmetric. The Jacobi identity telling us thatthere is a Lie algebra behind the Yang-Mills theory arises at the quartic level only.

The technical difficulty of a theory with at least one higher-spin field is that it is noteven possible to consistently consider an interaction of finite amount of fields, [6,17,31,32].Altogether, this makes Fronsdal program (construction of interacting theory for higher-spin gauge fields) extremely difficult to implement. At present, the traditional methodsof metric-like approach has led to little progress in this direction, basically their efficiencystops at the level of cubic interaction, see however [31, 33]. This state of affairs made itclear that some other tools were really relevant to push the problem forward.

A very fruitful direction initiated by Fradkin and Vasiliev and then largely extendedby Vasiliev that eventually foster the appearance of complete nonlinear system for higher-spin fields, [34–39], is the so called unfolded approach, [40, 41], to dynamical systems.It rests on the frame-like concept rather than the metric-like and the differential formlanguage which makes the whole formalism explicitly diffeomorphism invariant. This isthe branch of higher-spin theory that we want to discuss in this lectures. As we willsee the concept of gauge symmetry intrinsically resides in the unfolded approach and,therefore, it suits perfectly for analysis of gauge systems.

The other advantage is that being applied to a free system it reveals all its symmetriesand the spectrum of (auxiliary) fields these symmetries act linearly on. Particularly, thehigher-spin algebra is something that one can discover from free field theory analysisusing the unfolded machinery. This is one of the corner stones towards the nonlinearhigher-spin system. Nevertheless, the unfolded approach is just a tool that controls gaugesymmetries, degrees of freedom and coordinate independence while containing no extraphysical input. It turned out to be extremely efficient for higher-spin problem providingus with explicit nonlinear equations, still it gives no clue for their physical origin.

The Vasiliev equations, [34, 36, 39, 42, 43], are background independent. The AdSvacuum relevant for propagation of higher-spin gauge fields arises as some particularexact vacuum solution, while propagation around other vacuums have received no physicalinterpretation so far, neither the geometry of space-time is known. Another issue aboutthe unfolded formalism is its close relation to integrability. In its final form the space-timeequations acquire zero-curvature condition which states that space-time dependence getsreconstructed from a given point in a pure gauge manner. Although in principle anydynamical system can be put into the unfolded form, in practice it is rarely possible todo so explicitly. Surprisingly, the unfolded form of unconstrained Yang-Mills (s = 1) andgravity (s = 2) in four dimensions is still not known unlike the complete theory of all

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massless spins.We consummate these lectures with the equations of motion for nonlinear higher-

spin (HS) bosonic fields which are known in any space-time dimension. A substantialprogress in these theories has been achieved in lower dimensions, d = 3 and d = 4. Theseare the dimensions where the two-component spinor formalism is available. Not only itsimplifies formulation of the equations, it as well reduces technical difficulties in theiranalysis drastically. We restrict ourselves to the simplest case of four dimensional bosonicsystem in these notes. The main reason why this simplification really takes place is due tounconstrained spinorial realization of HS algebras accessible in lower dimensions. To givean idea how it works, we briefly touch on HS algebras. The effect of spinorial realizationis to a large extent similar to the one in General Relativity in four dimensions when usingNewman-Penrose formalism.

There are plenty of issues of higher-spin theory left untouched in this review. Partic-ularly, we do not discuss questions on the structure of higher-spin cubic vertices, stringinspired, [44–46], and BRST-type formulations [47]. We totally left aside issues relatedto alternative parent type formalisms, [48–50], developing side by side with the unfoldedapproach as being still much less purposeful for interacting theories. And even withinthe Vasiliev theory there are a lot of interesting problems that we consciously evade.Among those are problems of AdS/CFT correspondence, [51–54], and related subjects ofexact solutions, [55–57]. This field has recently received great deal of attention especiallyin d = 3, [58–62], and develops rapidly. We believe it deserves a separate review, thebeginning of the story is in [63].

Our goal was to make the reader familiar with the apparatus of unfolding approachwhich sometimes gives an impression as being bordered on tautology with unexpectedpower for unconstrained spinorial systems like those in three and four dimensions. Fi-nally and most importantly we wanted to introduce the Vasiliev nonlinear equations andthe technique to operate with them. In our own perception of the field the absence ofunderlying physical grounds and strictly speaking the absence of derivation of the equa-tions themselves have always been a great source of confusion. It made us wonder of anysmooth way of presenting the subject. In writing this self-contained and non-technicalreview we tried to emphasize the issues we had problems with ourselves while studyinghigher-spin theory. The review looks quite lengthy, but it is the price we pay for beingelementary and we believe that some parts can be dropped depending on the reader’sbackground.

There is a number of reviews devoted to different aspects of higher-spin theory, [38,64–67]. These notes are based on the lectures on Vasiliev higher-spin theories given bythe authors at the Galileo Galilei Institute, Florence. We appreciate any comments,suggestions on the structure of the lectures as well as pointing out missing references.Please feel free to contact us in case you have any questions or found some places difficultto understand.

The outline of these notes is as follows. We begin with two sections that are not directlyrelated to the Vasiliev higher-spin theory — the review of the Fronsdal formulation andintroduction to the frame-like formulation of gravity. The logic of the rest of the sectionsis first to convert the Fronsdal theory into the unfolded form, Section 4 for the gravity caseand Section 5 for arbitrary spin fields. Once a number of unfolded examples is collected

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we turn to a more abstract description of what the unfolded approach is, Section 6,where we emphasize its relation to Lie algebras and representation theory. The reasonwhy lower dimensions are more tractable is thanks to exceptional isomorphisms, of whichwe need so(3, 1) ∼ sl(2,C). The dictionary for tensors of so(3, 1) and spin-tensors ofsl(2,C) is explained in Section 8. In Section 9 using the vector-spinor dictionary wereformulate the unfolded equations found for any d in Section 5, which allows one toswitch the cosmological constant on easily. The AdS4 unfolded equations for all spinsalready contain certain remnants of higher-spin algebra, which is discussed in Section 10.With all ingredients being available we proceed to Vasiliev equations in Section 11.

There are also a number of extra sections, e.g. the one devoted to the MacDowell-Mansouri-Stelle-West formulation of gravity, which are not necessarily needed to proceedto Vasiliev equations. Other extra sections are devoted to more advanced topics. Thereis also a number of appendices containing our index conventions and an introduction toYoung diagrams and tensors which is of some importance since the higher-spin theory isfirst of all a theory of arbitrary rank tensors.

2 Metric-like formulation for free HS fields

In this section we collect some useful facts about metric-like description of spin-s fields,[1,68–70]. Following traditional field theory the subject is pretty standard and have beenreviewed many times, see e.g. [64,71,72]. For the sake of simplicity we deal with bosonicfields only. Description of fermions is in many respects qualitatively similar. Althoughinteracting massless higher spin fields are believed not to exist in flat space-time, whichis a subject of many no-go theorems, see e.g. [5] for a review, the free fields do – makingit useful firstly to discuss the case of spin-s fields in Minkowski space and then proceedwith (anti)-de Sitter.

2.1 Massless fields on Minkowski background

Standard way of thinking of massless fields is as those that admit local gauge invarianceresponsible for a reduced number of physical degrees of freedom as compared to nongauge,massive, fields. This is generally true except for matter fields, s = 0, 1/2 which beingnongauge still can be massless. In Minkowski space-time a massless spin-zero field φ(x)is the one that obeys φ(x) = 0, i.e. have zero mass-like term. Spin-one is a gauge field1

that is described by a gauge potential Am(x) obeying Maxwell equation Am−∂m∂nAn =0 which remains invariant under local gauge transformation Am ∼ Am + ∂mξ. Theseare the well known examples of lower spin massless fields s = 0, 1 which, as we willsee, naturally fit the general spin-s free field description developed first by Fronsdal,[1, 68]. Still, these lower spin examples are to some extent degenerate exhibiting nospecial features characteristic for s ≥ 3.

The less degenerate case of spin-two field can be described by a symmetric tensorφmn = φnm with gauge symmetry δφmn = ∂mξn + ∂mξn. This can be easily achieved

1Lowercase Latin letters a, b, c, ... are for the indices in the flat space, which are raised and loweredwith ηmn = diag(−,+, ...,+).

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from the fully nonlinear classical theory of Einstein gravity through its linearization.To do so, one identifies φmn with the fluctuations gmn = ηmn + κφmn of the metricfield gmn over the Minkowski background ηmn. Here κ is a formal expansion parameter.The gauge symmetry δφmn = ∂mξn + ∂mξn comes about as linearized diffeomorphismδgmn = ∂mξ

cgcn + ∂nξcgcm + ξc∂cgmn. Indeed, to the lowest order we have

δgmn = δφmn = ∂mξcηcn + ∂nξ

cηcm = ∂mξn + ∂mξn . (2.1)

The equations for φmn can be obtained via linearization of the Einstein equations Rmn −12gmnR = 0 and coincide with the Fronsdal equations for s = 2, see below. Of course,

the free spin-two field can be defined without any reference to Einstein-Hilbert action,differential geometry and diffeomorphisms.

As it was shown by Fronsdal, [1,68], a massless spin-s field can be described by a totallysymmetric rank-s tensor field2 φa(s) ≡ φa1...as which obeys an unusual trace constraint

φa(s−4)bcdeηbcηde ≡ φa(s−4)mnmn ≡ 0 . (2.2)

Clearly, the trace constraint becomes effective starting from s = 4 being irrelevant fors = 0, 1, 2, 3. It tells us that the Fronsdal field consists of two irreducible (symmetric andtraceless) Lorentz tensors of ranks s and s − 2. Indeed, having an arbitrary symmetrictensor φm1...mk

one can always decompose it into a sum

φm(k) = φ′m(k) + ηmmφ

′′m(k−2) + ηmmηmmφ

′′′m(k−4) + . . . , (2.3)

where all ’primed’ fields are traceless. Eq. (2.2) states then that Fronsdal field φm(s) isonly allowed to have φ′

m(s) and φ′′m(s−2) to be non-zero. It is already this odd constraint

that puzzles and complicates things a lot at interacting level as it would be more naturalto work with fully unconstrained tensors or totally traceless ones. We will see in Section5 that (2.2) arises naturally from the extension of the frame-like formulation of gravity tofields of any spin.

The dynamical input is given by Fronsdal equations

F a(s) = φa(s) − ∂a∂mφma(s−1) + ∂a∂aφa(s−2)mm = 0 , (2.4)

which are invariant under the gauge transformations3

δφa(s) = ∂aξa(s−1) , ξa(s−3)mm ≡ 0 , (2.5)

where the gauge parameter is traceless (this becomes effective for s ≥ 3).This generalizes the cases of spin-0, 1, 2 to any s. The Fronsdal equations are valid

for s = 2 as well, coinciding in this case with the linearized Einstein equations. They arevalid for s = 1 and s = 0 if we notice that the third and the second terms are absent fors = 0, 1 and s = 0, respectively.

2Since the number of indices that a tensor can carry is now arbitrary we need a condensed notation.All indices in which a tensor is symmetric or needs to be symmetrized are denoted by the same letterand a group of s symmetric indices a1...as is abbreviated to a(s). The operator of symmetrization sumsover all necessary permutations only, e.g. V aUa ≡ V a1Ua2 + V a2Ua1 . More info is in Appendix B.

3The verification of this fact is in Appendix B.

8

The tracelessness of ξa(s−1) goes hand in hand with the double-tracelessness of theFronsdal field. Indeed, from (2.5) it follows that φa(s) should be double traceless onceξa(s−1) is traceless. The second trace of φa(s) is not affected by the gauge symmetry.In playing with trace constraints for field/gauge parameter one finds that the Fronsdaltheory is essentially unique4. To this effect it is instructive to look at the variation of theFronsdal operator under (2.5) with ξa(s−1) not satisfying any trace constraints

δF a(s) = (3∂a∂a∂a − 2ηaa∂a) ξa(s−3)mm . (2.6)

Going on-shell. In order to see that the solutions to the Fronsdal equation do carry aspin-s representation of the Poincare group it is useful to define de-Donder tensor

Da(s−1) = ∂mφma(s−1) − 1

2∂aφa(s−2)m

m , (2.7)

δDa(s−1) = ξa(s−1) , (2.8)

F a(s) = φa(s) − ∂aDa(s−1) , (2.9)

which transforms in a simple way under the gauge transformation and constitutes the non-φ part of the Fronsdal operator. Since the de-Donder tensor carries as many componentsas the gauge parameter (2.8) it is possible to gauge it away Da(s−1) = 0. One is left withthe gauge transformations ξa(s−1) obeying ξa(s−1) = 0. Since ξa(s−1) can be nonzero onecan further impose one more condition, φa(s−2)m

m = 0. Indeed, φa(s) is now on-shell, i.e.φa(s) = 0, and δφa(s−2)m

m = 2∂mξma(s−2). The gauge-fixed equations and constraints

read

φa(s) = 0 , ξa(s−1) = 0 , (2.10)

∂mφma(s−1) = 0 , ∂mξ

ma(s−2) = 0 , (2.11)

φa(s−2)mm = 0 , ξa(s−3)m

m = 0 , (2.12)

δφa(s) = ∂aξa(s−1) . (2.13)

These are all the Lorentz covariant conditions one can impose without trivializing thesolution space. There is still a leftover on-shell gauge symmetry, which manifests inde-composable structure of the representation carried by φa(s).

4In principal one can relax the tracelessness of the gauge parameter preserving double-tracelessness ofthe field. However, as is seen from (2.5), this would result in differential constraint on a gauge parameterwhich is not a safe thing for it typically affects the number of physical degrees of freedom and havinga differential constraints on fields/gauge parameters complicates the study of interactions a lot. If thefield were irreducible, i.e. it is symmetric and traceless, one would derive ∂mξa(s−2)m = 0 by takingtrace of (2.5), which is again a differential constraint. Imposing such a constraint makes sense at thefully nonlinear level in the case of s = 2, which corresponds to volume-preserving diffeomorphisms, seee.g. [73]. The generalization to the spin-s case is also possible, [72, 74]. Alternatively, one could try toproject ∂aξa(s−1) onto the traceless component, in which case one would find no gauge invariant equations,so this option is unacceptable.

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Counting degrees of freedom. To count degrees of freedom5 one solves (2.10) byperforming the Fourier transform

φa(s)(x) =

∫ddp δ(p2)ϕa(s)(p)eipx (2.14)

and analogously for ξa(s−1).Since the equations are Lorentz covariant all points in momentum space are equivalent

and we can look at any pm, pmpm = 0 to count the number of independent functions. A

convenient choice is given by light-cone coordinates

x± =1√2(x1 ± x0) , xi = x2, ..., xd−1 . (2.15)

Indices range a = +,−, i, i = 1...(d − 2) in these coordinates with the metric η+− =η−+ = 1, ηij = δij and all other components being zero. Let us take pm = eδ+m, where e issome constant and δ+m is the Kronecker delta. The components of ϕa(s) can be split into

ϕ

N︷ ︸︸ ︷+...+

M︷ ︸︸ ︷−...− i(s−M−N) . (2.16)

Eq. (2.11) tells that all components with at least one + direction vanish, ϕ+a(s−1) = 0(analogously for ξa(s−1)). Now one can use gauge symmetry (2.13) δϕa(s) = eη+aξa(s−1),i.e. δϕ−a(s−1) = eη+−ξa(s−1) to set ϕ−...−i...i = 0. Finally, we are left with ϕi(s)(p) thatis symmetric and so(d − 2)-traceless, i.e. traceless with respect to δij. Indeed, ϕa(s) isso(d− 1, 1)-traceless, which can be rewritten as

ϕa(s−2)mm ≡ ϕa(s−2)ijδij + 2ϕa(s−2)+−η+− ≡ 0 . (2.17)

Then, we note that the last term carries at least one index along ’+’-direction, so it iszero by (2.11). To conclude, the degrees of freedom are those of an irreducible rank-sso(d−2)-tensor times the dependence on pm that lives on (d−1)-dimensional cone. Thatϕi(s)(p) is an irreducible rank-s tensor at each p, i.e. symmetric and traceless, impliesthat it describes a single spin-s particle.

Lagrangian. The Fronsdal Lagrangian reads

S = −12

∫

Md

(∂mφ

a(s)∂mφa(s) −s(s− 1)

2∂mφ

n a(s−2)n ∂mφ k

k a(s−2) +

+ s(s− 1)∂mφn a(s−2)n ∂kφkma(s−2) − s∂mφma(s−1)∂nφna(s−1)+ (2.18)

−s(s− 1)(s− 2)

4∂mφ

n ma(s−3)n ∂rφ k

r ka(s−3)

).

5One would like to prove that the solution space carries a unitary irreducible representation of thePoincare group. A representation corresponding to massless particle can be shown to be induced froma finite-dimensional representation of the Wigner’s little algebra so(d − 2). It is the representations ofso(d − 2) that define the spin. See, e.g. [75] or the second chapter of the Weinberg’s QFT textbook, forthe review of the Wigner’s construction.

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It is fixed up to an overall factor and total derivatives by the gauge symmetry, (2.5), [76].It can be put into a more compact form by integrating by parts

S =1

2

∫

Md

φa(s)Ga(s) , Ga(s) = F a(s) − 1

2ηaaF a(s−2)m

m , (2.19)

where the trace of the Fronsdal operator is

F [φ]a(s−2)mm = 2φa(s−2)m

m − 2∂n∂mφmna(s−2) + ∂a∂mφ

a(s−3)mnn . (2.20)

The gauge invariance of the action implies the Bianchi identities

0 = δS = s

∫

Md

∂bξa(s−1)Ga(s−1)b = −s

∫

Md

ξa(s−1) ∂mGa(s−1)m = −s

∫

Md

ξa(s−1)Ba(s−1) ,

i.e. the following linear operator annihilates the equations of motion6

B[F ]a(s−1) = ∂mFa(s−1)m − 1

2∂aF a(s−2)m

m , B[F [φ]] ≡0 . (2.21)

Again, it is instructive to see how the Bianchi identities get violated if φ does not obeythe double-trace constraint,

B[F [φ]]a(s−1) =

(−32∂a∂a∂a + ηaa∂a

)φa(s−4)mn

mn . (2.22)

Let us note that the equations that come from Lagrangian, (2.19) or (2.18), is Ga(s) = 0and is a little bit different from (2.4). They are equivalent in fact. Indeed, taking thetrace

Ga(s−2)mm = −d + 2s− 6

2F a(s−2)m

m , (2.23)

we see that Ga(s) = 0 implies Ga(s−2)mm = 0 and hence F a(s−2)m

m = 0. On substitutingthis back to Ga(s) = 0 one finds F a(s) = 0. It was important that both Ga(s) and F a(s) aredouble traceless as a consequence of φa(s−4)mn

mn ≡ 0.

The long and winding road from representations to Lagrangians. It is worthstressing that a systematic approach to Lagrangians can be quite difficult requiring toanswer a priori four different questions.

(i) to classify all unitary irreducible representations of the space-time symmetry group,Poincare in our case. Postulates of Quantum Mechanics combined with the Special Rel-ativity (i.e. the idea that the physical laws are covariant under the Poincare algebra)results in the statement that all systems, e.g. particles, must carry a unitary representa-tion of the Poincare algebra, [77]. This is where the notion of spin and mass comes outas parameters specifying a representation. The representation theory of Poincare algebrahas little to do with the space-time directly.

6Let us note that ∂mGa(s−1)m has one more term, − 12η

aa∂mF a(s−3)mnn , which is projected out thanks

to traceless ξa(s−1), which again shows the importance of ξa(s−3)mm ≡ 0.

11

(ii) to realize these representations on the solutions of certain P.D.E’s imposed oncertain tensor fields over the Minkowski space. We have seen that a spin-s representationis realized on ϕi(s)(p) where p2 = 0 and has again a little to do with the space-time. An on-shell description is given by (2.10)-(2.13) in terms of φa(s)(x) that is defined up to a gaugetransformation. At this stage we see that ϕi(s) comes as projection/factor of ϕa(s) and inprinciple one can imagine embedding ϕi(s) into an so(d−1, 1)-tensor with more indices suchthat the equations/gauge symmetries project out redundant components. The numberof indices that a field may carry is not directly related to the spin as a parameter ofan irreducible representation, one has to take equations/gauge symmetries into account.There are generally infinitely many descriptions of one and the same representation bydifferent combinations of field/P.D.E./gauge-symmetry. The simplest example is a spin-one particle, photon, which can be equally well described by gauge potential Am, Am−∂m∂

nAn = 0, δAm = ∂mξ or by nongauge field strength Fmn = −Fnm, ∂nFmn = 0,

∂aFbc + ∂bFca + ∂cFab = 0. There is a generalization of this example to fields of all spinshigher than one, which is in Section 5.5.

(iii) to find an off-shell description, i.e. to extend fields/gauge parameters in such away that no differential constraints like (2.10)-(2.12) remain. An off-shell description aswe have seen requires adding a traceless rank-(s − 2) tensor to the traceless φa(s) to becombined together into a double traceless Fronsdal field.

(iv) to get these P.D.E.’s (or equivalent to them) as variational equations for certainLagrangian. Coincidentally, in the case of massless spin-s fields in Minkowski or anti-deSitter space the same field content as we used for an off-shell description is sufficient towrite down a Lagrangian, which is not true for the massive spin-s field, [78, 79].

2.2 Massless fields on (anti)-de Sitter background

The Fronsdal theory can be easily extended to the constant curvature background, [69,70],which are the maximally symmetric solutions of Einstein equations with cosmologicalconstant Λ. These are known as de Sitter, Λ > 0, and anti-de Sitter, Λ < 0 spaces. Thealgebraic double trace constraint (2.2) remains unchanged but ηmn gets replaced with7

the (anti)-de Sitter metric8 gmn(x)

φm(s−4)nnrr gnngrr ≡ 0 . (2.25)

All derivatives should be covariantized and we use the normalization

[Dm, Dn]Va = ΛδamgnbV

b − ΛδangmbVb (2.26)

7From now on we reserve lowercase Latin letters a, b, c, ... for the indices in the flat space, which areraised and lowered with ηmn. Underlined lowercase Latin letters a,m, n, r, k, ... are the ’world’ indicesbeing raised and lowered with some generally nonconstant metric gmn.

8Equivalently we can transfer all indices to the fiber with the help of the tetrad/frame/vielbein fieldham, which will be introduced systematically in the next Section. In fact, it has to be introduced once we

would like to include fermions. Then the algebraic constraints do not change at all

φa(s−4)bbccηbbηcc ≡ 0 , φa(s) = φm(s)hma...hma . (2.24)

In taking derivatives we can use [Dm, Dn]Va(s) = Λha

mhnbVba(s−1) − Λha

nhmbVba(s−1).

12

and Λ is the cosmological constant. The gauge transformation law now reads

δφa(s) = ∇aξa(s−1) . (2.27)

The Fronsdal operator is promoted to

F a(s)[φ] = φa(s) −∇a∇mφma(s−1) +

1

2∇a∇aφa(s−2)m

m −m2φφ

a(s) + 2Λgaaφa(s−2)mm ,

m2φ = −Λ((s− 2)(d+ s− 3)− s) , (2.28)

where we note the appearance of the mass-like terms. Mind that ∂a∂a in the third termhas changed to 1

2∇a∇a since covariant derivatives do not commute and s(s−1) terms are

now needed to symmetrize a(s) as contrast to s(s− 1)/2 terms in flat space. In checkingthe gauge invariance of the equations we find that the leading terms vanish thanks tothe gauge invariance of the Fronsdal operator in flat space. However, in order to cancelthe gauge variation we have to commute some of the derivatives. The commutatorsproduce certain new terms that can fortunately be canceled by adding mass-like terms.The strange value of m2

φ can be derived using the representation theory of so(d− 1, 2) orso(d, 1), [80, 81] which are the symmetry algebras of anti-de Sitter and de Sitter spaces,respectively. It is related to the Casimir operator in the corresponding representation.We will have more to say about (anti)-de Sitter later on. The lesson is that masslessnessthat imply gauge invariance does not necessarily imply the absence of mass-like terms.Protection of gauge invariance is more important than the presense/absence of mass-liketerms as it guarantees that the number of physical degrees of freedom does not changewhen switching on the cosmological constant. The constant curvature of (anti)-de Sitterspace acts effectively as a harmonic potential that renormalizes the value of the massterm. Let us also note, that the precise value of the mass-like term depends also on theway the covariant derivatives in the second term of (2.28) are organized.

The action has formally the same form

S =1

2

∫φa(s)G

a(s) , Ga(s) = F a(s) − 1

2gaaF a(s−2)m

m , (2.29)

where the trace of the Fronsdal operator reads

F [φ]a(s−2)mm = 2φa(s−2)m

m − 2∇n∇mφmna(s−2) +∇a∇mφ

a(s−3)mnn −m2

1φa(s−2)m

m ,

m21 = −2Λ(s− 1)(d+ s− 3) . (2.30)

Finally let us note that (2.6) and (2.22) are still valid upon replacing ∂a → ∇a.The process of imposing gauges is analogous to the case of the Minkowski background.

Given this, without going into details, we can conclude that equation (2.30) describes thesame number of physical degrees of freedom since it preserves the same amount of gaugesymmetry and the order of equations and Bianchi identities remains unchanged.

It is worth stressing that by going from Minkowski to more general backgrounds onecan lose certain amount of physical interpretation. For example, in anti-de Sitter spacethe space-time translations, Pa, do not commute so one cannot diagonalize all Pa simul-taneously. In particular, PaP

a is no longer a Casimir operator as it is the case in the

13

Minkowski space. Nevertheless, in some sense anti-de Sitter algebra so(d− 1, 2) is betterthan Poincare one in being one of the classical Lie algebras, while Poincare algebra isnot semi-simple, which leads to certain peculiarities in constructing representations of thelatter. The Poincare algebra can be viewed as a contraction of so(d− 1, 2). The particlesin the case of anti-de Sitter algebra so(d− 1, 2) should be defined as Verma modules withspin and mass being related to the weights of so(d − 1, 2), which is somewhat technicaland we refer to [80, 82, 83].

As for the field description the best one can do on a general background is to ensurethat the number and order of gauge symmetries/equations/Bianchi identities remainsunchanged (or get changed in a coherent way) so as to preserve the number of degrees offreedom, [84].

Once the gravity is dynamical or the background is different from (anti)-de Sitteror Minkowski, the Fronsdal operator is no longer gauge invariant. Indeed, in verifyingthe gauge invariance we have to commute covariant derivatives ∇’s. For the case of theMinkowski space ∇’s just commute. For the case of (anti)-de Sitter (constant curvature)space the commutator is proportional to the background metric, so the commutatorsproduce mass-like terms. In generic background we are left with

δF = R...∇ξ... 6= 0 (2.31)

where R... is the Riemann tensor. Therefore the Fronsdal operator becomes inconsistenton more general configurations of metric.

In particular when the metric gmn becomes a dynamical field we face the problem ofhow to make higher-spin fields interact with gravity. This was the starting point for theno-go [85] by Deser and Aragone and then yes-go results by Fradkin and Vasiliev [3, 4](see review [5] on various no-go theorems related to higher-spins). Some comments on theFronsdal theory on general Riemannian manifolds can be found in extra Section 12.1.

We see that there is something special about higher-spin fields, the threshold beings = 2, since all lower-spin fields, s = 0, 1

2, 1 can propagate on any background, gmn, and

the graviton is self-consistent on any background of its own.

Summary. There is a well-defined theory of free fields of any spin-s on the specificbackgrounds, which are Minkowski and (anti)-de Sitter — maximally symmetric solutionsof Einstein equations with cosmological constant. The fields and gauge parameters haveto obey certain trace constraints, (2.2), (2.5.b).

3 Gravity as gauge theory

Among theories of fundamental interactions there are Yang-Mills gauge theories basedon (non)abelian Lie algebras and General Relativity (GR) that stands far aside and istypically viewed as essentially different from gauge theories. Particularly, the way it wasformulated by Einstein, GR does not rest on any gauge group. On the other hand, gravityclearly has a gauge symmetry represented by arbitrary coordinate transformations anddiffeomorphisms. From that perspective it seems quite natural to address a question of agauge form of GR.

14

This section is aimed to demonstrate that gravity can in many respects be thoughtof as a gauge theory. The relevant variables to see this are the so called vielbein eam and

spin-connection ωa,bm , which can to some extent be treated as components of a Yang-Mills

connection of Poincare, iso(d − 1, 1), de Sitter, so(d, 1), or anti-de Sitter, so(d − 1, 2),algebras. The reader familiar with Cartan formulation of gravity can skip the entiresection. We begin with a very short and elementary introduction to the Cartan geometry,then proceed to various ways of thinking of gravity as a gauge theory. The MacDowell-Mansouri-Stelle-West formulation of gravity is left to the extra Section 12.2. The relevantreferences include [86–89] and [90] for the references on the original papers by Cartan,Weyl, Sciama, Kibble.

3.1 Tetrad, Vielbein, Frame, Vierbein,....

In differential geometry one deals with manifolds – something that can be built up fromseveral copies of the Euclidian space. The point is that not every hyper-surface we canimagine is homeomorphic to a Euclidian space and hence can be covered by some globalcoordinates. Therefore we have to cut a generic manifold into smaller overlapping pieceseach of which can be thought of as a copy of Euclidian space. We need transition functionsthat allow one to identify the regions of two copies of Euclidian space whose images overlapon the manifold. A manifold itself then comes as a number of copies of Euclidian space(patches) together with the transition functions that are defined for certain pairs of copiesand obey certain consistency relations.

In differential geometry framework the objects, tensors, one uses transform properlyunder the change of the coordinates so that the scalar (physical) quantities we computedo not depend on the choice of coordinates. Despite the fact that differential geometryis designed in a democratic way with respect to different coordinates, this is not fully sofor tensors. Indeed, given a tensor T its components Tm...

n... are given with respect tothe basis in the tangent space that is induced from the coordinates in the current chart.The basis vectors at a given point are vectors that are tangent to the coordinate lines,see the figure. We will refer to such ’bare’ tensors as to world tensors and to the indicesthey carry, m,n, ..., as to world indices. To disentangle the basis in the chart and in thetangent space we may introduce an auxiliary nondegenerate matrix eam(x) that transferstensor indices from the basis induced by the particular coordinates to some other basisin the tangent space we might prefer more. With the help of eam(x) each world tensoracquires an avatar

Tm...n... −→ T a...

b... = eam... Tm...

n... (e−1)nb ... (3.1)

and we refer to the tensor in the new basis as to the fiber (tangent) tensor and to theindices it carries, a, b, ... as to fiber (tangent) indices. In principle, tensors of mixed type,i.e. those that carry both world and fiber indices simultaneously are possible and suchtensors do appear in our study. But the rule of course is that only indices of the sametype can be contracted with either δab or δmn .

If no derivatives are around it is obvious that one can use either of the bases foralgebraic computations, e.g. taking tensors products or contracting indices, i.e. the

15

Figure 1: The basis vectors in the tangent plane TMO at some point O are by definitionthe vectors that are tangent to the coordinate lines. Let p(x1, ..., xn) be the point on themanifold parameterized by Cartesian coordinates (x1, ..., xn) in some chart, we can thinkof it as a point in a bigger Euclidian space where the given manifold is embedded. Then,~ei =

ddtp(x1, ..., xi + t, ..., xn)|t=0

~e2

~e1

O

TMO

BA

p(x1, ..., xn)

x2

x1

O B

A

following diagram commutes

set of world tensors Tm...n... , ...

e,e−1

−−−→ set of fiber tensors T a...b... , ...yoperations

yoperations

derived set of world tensors T n...n... , ...

e,e−1

−−−→ derived set of fiber tensors T a...a...

(3.2)

In other words, having a set of world tensors first, we can either do some algebraiccomputations like taking tensor products or contracting the indices and then transfer allindices left free into the fiber ones with the help of eam, (e

−1)ma or we can first transfer allthe indices to fiber ones and then perform identical computations but in the fiber.

There is a strong motivation from physics to introduce eam — the equivalence prin-ciple. In the famous Einstein’s thought experiment an experimentalist, when put intoa free falling elevator without windows, cannot tell whether she is falling freely in thegravitational field or is left abandoned in the open space far away from any sources ofgravitational field9. This has led Einstein to the equivalence principle (EP). EP impliesthat locally one can always eliminate the gravitational field by taking a freely falling ele-vator. This statement lies at the core of all problems in defining stress-tensor of gravity.As we are going to consider gravity, there is a preferred set of bases given by Einstein’selevators — elevators that are freely falling in the local gravitational field — the physicsin this elevator is locally as in the Special Relativity (SR). The latter is true for physicallysmall elevators, i.e. up to tidal forces, etc.

9Equivalently, gravitational field is locally indistinguishable from the accelerating frame.

16

The EP tells us that the metric in the new basis, which is associated with the elevator,is constant, for example, ηab = diag(−++...+), i.e. we have

ηab = eam(x) gmn(x) ebn(x) , η = eT g e , (3.3)

or, equivalently, one can always recover the original metric10 gmn

gmn = eam(x) ηab ebn(x) . (3.4)

The object eam(x), i.e. the Einstein’s elevator, is called tetrad or vierbein in the case offour dimensional space-time; vielbein, soldering form or frame in arbitrary d; zweibein,dreibein, etc in case of two, three, etc. dimensions.

It is worth noting that the metric as a function of the vielbein is defined in such a waythat different ways of raising and lowering indices lead to the same result. For example,the inverse vielbein ema is just the matrix inverse of eam, but it can also be viewed as eamwhose indices were raised/lowered with gmn and ηab,

ema = (e−1)ma = gmn ebn ηba . (3.5)

Obviously, eam(x), being a d × d matrix that depends on x, has enough componentsto guarantee (3.3). A change of coordinates xm → ym amounts to defining d functionsym = fm(xn), i.e. it has less ’degrees of freedom’ as compared to the vielbein. Indeed, inorder for eam(x) to be equivalent to a change of coordinates it must be eam = ∂mf

a(x). Theintegrability of this condition, i.e. 0 ≡ (∂n∂m−∂m∂n)fa(x), implies ∂me

an−∂neam = 0, which

is generically not true. It is obvious that one cannot remove gravitational field everywherejust by a coordinate transformation since there are tensor quantities like Riemann tensorRmn,rk and Rmn,rk = 0 is a coordinate independent statement.

The equivalence principle leads to an idea of General Relativity (GR) as being alocalization (gauging) of Special Relativity (SR) and this is the idea we would like to followand to generalize to fields of all spins. SR can be thought of as the theory of the globalPoincare invariance, i.e. a theory of ISO(d− 1, 1) as a rigid symmetry. Which amount ofthis symmetry gets localized in GR? Apparently the elevators form an equivalence classsince given an elevator eam(x) one can rotate it and boost it at any velocity v. Thesetransformations belong to the Lorentz group SO(d − 1, 1). At each point (or physicallyspeaking at small neighborhood of each point) we have a different elevator and hence theLorentz transformations can depend on x. To put it formally, eam(x) and

e′am(x) = Aab (x)e

bm(x) , (3.6)

where A(x) ∈ SO(d − 1, 1), i.e. ATηA = η, produce the same gmn and does not changeηab. If the transformation is small, i.e. Aa

b is close to the unit matrix, we can writeAa

b = δab − ǫa,b and ǫa,b ≡ ǫac ηcb is antisymmetric, ǫa,b = −ǫb,a. Then a small change in

eam results in

δeam(x) = −ǫa,b (x)ebm(x) , (3.7)

which is a localized version of (3.6).

10If we forget about the x dependence, eam is a matrix that diagonalizes the given quadratic form gmn.

17

However, we lost translations of ISO(d − 1, 1) as the local symmetry. Translationbrings an elevator to another point where the gravitational field may differ. As we willsee local translations are not genuine symmetries.

Now the metric gmn can be viewed as a derived object and not as a fundamental. Everystatement in the language of gmn can be always rewritten in the language of eam and notvice verse because eam is defined up to an x-dependent Lorentz rotation in accordancewith the fact the Einstein’s elevator is not unique. We can also see this by countingindependent components, gmn has d(d + 1)/2 components, while eam has d2 components.The Lorentz transformations form a d(d− 1)/2-dimensional group, so

#vielbein −#Lorentz = #metric , (3.8)

which means that we did not lose or gain any new ’degrees of freedom’.There is one more fundamental reason to introduce the vielbein — matter fields, e.g.

electrons, protons, neutrons, which are fermions and thus are represented by spinors. Theystill do experience gravitational interaction and we have to deal with this experimentalfact. Let us emphasize that the very definition of spinors relies on the representation the-ory of the Lorentz algebra so(d−1, 1), which in the Minkowski space of Special Relativityis a subalgebra of the full Poincare symmetry algebra iso(d−1, 1). The notion of spin andmass rests on the representation theory of iso(d−1, 1) too. These are the parameters thatdefine unitary irreducible representations of iso(d− 1, 1). The existence of spinors, whichis due to the first homotopy group of SO(d− 1, 1) being nontrivial, makes it possible toconsider the action of the group up to a phase which distinguishes between contractibleand non-contractible paths on the group. This leads to a bizarre consequences, e.g. theelectron wave function changes its sign upon 2π-rotation.

A theory formulated in terms of some tensor representations of the Lorentz algebra,which are then used to define tensor fields over the Minkowski space, can be straightfor-wardly extended to a theory that has gl(d) as a symmetry algebra and then to a diffeo-morphism invariant theory. Clearly, having an so(d − 1, 1)-tensor T abc... in some theorywe can replace it with a tensor of gl(d) of the same type. Then, having a tensor of gl(d)we can turn it into a field T abc...(x) and make it transform under diffeomorphisms, see thetable below for some examples. However, there is no straightforward lift of spin-tensorrepresentations of so(d − 1, 1) to gl(d). Apparently11 we do not know of what replacesspin-tensors in the case of gl(d). The vielbein solves this problem as we can put ourselvesinto the reference frame where the symmetry algebra is so(d− 1, 1), the difference is thatit is a local statement. In order to construct Lagrangians and field equations we need toextend the covariant derivative to tensors with fiber indices.

Covariant derivative. It is necessary to define the covariant derivative in the fiber,then we can make it act in any representation of the Lorentz algebra, so(d − 1, 1), in

11Formally the fundamental group of SL(d), GL(d) = SL(d)×GL(1), is the same as for SO(d), becauseit is determined by the maximal compact subgroup. However, the double-valued representations of SL(d)are infinite-dimensional. A possible way out is to take infinite-dimensional spinorial representations ofSL(d) seriously, [91]. Such representations, when restricted to SO(d), decompose into an infinite sumof spin-tensor representations of all spins and hence contain higher-spin fields. As we will learn theconsistency of higher-spin theory requires infinite number of higher-spin fields, so at the end of the daySL(d)-spinors may not be so far away, [92].

18

SR GR

general Lagrangian∫ddxL(φ, ∂mφ, ηab)

∫ √|g| ddxL(φ,∇mφ, g

mn)

spin-zero 12

∫ddx ∂aφ ∂bφ η

ab 12

∫ √|g| ddx ∂mφ ∂nφ gmn

spin-one 14

∫ddxFab F

ab 14

∫ √|g| ddxFmn Frk g

mrgnk

spin-half∫ddx ψγa∂aψ ???, wait for (3.27)

particular on spin-tensors and hence be able to write down the Dirac Lagrangian in thegravitational field. The covariant derivative needs to be defined in a way that the followingdiagram commutes, otherwise there will be too many problems in comparing the resultsof differentiation in the two bases (we still think that the simple recipe to replace ∂ withD = ∂ + Γ works well for world tensors so we do not want to abandon this knowledge),

Vmema−−−→ VayDn

yDn

DnVmema−−−→ DnVa

(3.9)

The diagram implies that we can first differentiate a tensor, then transfer it to anotherbasis, or first transfer it to another basis and then differentiate. The results must coincide.Since in the world basis a vector in two coordinate frames can be related by any GL(d)matrix the Christoffel symbol Γm

nr is a generic matrix in m, r. In the fiber basis any changeof coordinates must be a Lorentz transformation. For the same reason that we used tointroduce Γ we introduce the spin-connection ωn

a,b . It has two types of indices, the world

index is due to Dn and the two fiber indices makes it a matrix in the fiber. Inside thecovariant derivative each fiber index acts upon by the spin connection and each worldindex by the Christoffel symbol, e.g.

DnVm = ∂nVm + ΓrnmVr , DnV

m = ∂nVm − Γm

nrVr , (3.10)

DnVa = ∂nV

a + ωna,b V

b , (3.11)

DnTabc...m... = ∂nT

abc...m... + ωn

a,u T

ubc...m... + ωn

b,u T

auc...m... + Γr

nmTabcr + ... . (3.12)

Since only Lorentz rotations are allowed in the fiber we must have ωma,b = −ωm

b,a, wherewe have used the right to raise and lower fiber indices with the help of ηab. Equivalentlywe can impose Dmη

ab = 0 to find ωma,b antisymmetric. The consistency condition, the

condition for the diagram to commute, leads to

eamDnVa = DnVm , Vm = eamVa . (3.13)

Since this must hold for any Vm we get

Dneam = ∂ne

am + Γr

nmear + ωn

a,b e

bm = 0 . (3.14)

This is called the vielbein postulate. Several comments can be made about the postulate

19

• The vielbein postulate is analogous to Dmgnr = 0 postulate in that it is designed todisentangle algebraic manipulations with the help of e (or g) and covariant deriva-tives, i.e. it ensures that contractions of indices commute with covariant derivatives.Note that (3.14) induces Dmgnr = 0.

• (3.14) can be solved both for Γ and ω as functions of e and its first derivatives, seeAppendix C. This is supported by comparing the number of equations d3 with thetotal number of components of #Γ = d × d(d + 1)/2 and #ω = d × d(d − 1)/2,#Γ +#ω = #eqs.

• In the solution Γ(e) the vielbein comes all the way in combinations that can berecognized as g and ∂g. One recovers the usual Christoffel symbols.

• On the contrary, the solution ω(e) cannot be rewritten in terms of the metric gmn,which supports the vielbein being a fundamental field.

If we anti-symmetrize in (3.14) over mn and use that Γrmn is symmetric, we find

T anm = ∂ne

am − ∂mean + ωn

a,b e

bm − ωm

a,b e

bn = 0 , (3.15)

i.e. Γ disappears and the system of equations turns out to have a triangular form. We canfirst solve for ω and then for Γ. Explicit solution for ω is given in Appendix C. In casethere is no need for Γ we can use (3.15). It can be more compactly rewritten if we hidethe world indices by saying that eam and ωm

a,b are differential forms. A short introduction

to the language of differential forms can be found in Appendix D.Thinking of ean and ωn

a,b as degree-one differential forms, ea = dxn ean, ω

a,b = dxn ωn

a,b

one can rewrite (3.15) as

T a = dea + ωa,b ∧ eb = Dea = 0 . (3.16)

Two-form T a ≡ 12T amndx

m ∧ dxn is called the torsion. We can check the integrability of(3.16) applying d to (3.16) and using that d2 ≡ 0 and then using (3.16) again to expressdea. We have nothing to say on how dωa,b looks like so we keep it as it is. The result12 is

F a,b ∧ eb = 0 , F a,

b = dωa,b + ωa,

c ∧ ωc,b . (3.17)

The two-form F a,b has four indices in total and is in fact related to the Riemann tensor

Rmn,ru = Fmna,b ear e

bu . (3.18)

It is a painful computation to solve T a = 0 for ωa,b as a function of ea and then computeF a,b to see that ea appears in combinations that can be rewritten in terms of the metric.Fortunately, there is a back-door. Let us compute the commutator of two covariantderivatives on some vector V m and the same for V a = V meam, i.e. [Dm, Dn]V

r and[Dm, Dn]V

a. The two results must match after transferring all the indices to fiber ones or

12In its simplest form this is just the Frobenius integrability condition. Given a set of PDE’s ∂µφ(x) =fµ(x) the commutativity of partial derivatives imply 0 ≡ (∂µ∂ν − ∂ν∂µ)φ(x) = ∂µfν − ∂µfν = 0. The lastequality does not hold for a generic vector-function fµ, which means that the system can be inconsistent.

20

to world ones. We already know that [Dm, Dn]Vr is expressed in terms of the Riemann

tensor. Analogously, [Dm, Dn]Va can be expressed in terms of Fmn

a,b , which gives

Rmnru V

u =(Fmn

ab V

b)era =

(Fmn

ab e

buV

u)era =

(Fmn

ab e

bue

ra

)V u . (3.19)

The identity (3.17) can be then recognized as the first Bianchi identity for the Riemanntensor, it being a three-form anti-symmetrizes over the three indices in square brackets,

R[mn,r]u ≡ Rmn,ru + Rnr,mu +Rrm,nu ≡ 0 . (3.20)

Since everything in the metric-like formulation can be derived from the frame-like one, itis not surprising that the Einstein-Hilbert action13

SEH =

∫ √det g R (3.21)

can be rewritten in Cartan-Weyl form14

SCW =

∫F a,b(ω(e)) ∧ ec ∧ ... ∧ eu ǫabc...u . (3.22)

The integrand is a top-form, i.e. the form of maximal degree, which is the space-timedimension, and can be integrated. Let us note that ω used in the action is assumed to beexpressed in term of e via the vielbein postulate, (3.14), or the torsion constraint, (3.16),which obscures the interpretation of ω as a gauge field of the Lorentz algebra. This iswhat we would like to improve on.

3.2 Gravity as a gauge theory

Short summary on Yang-Mills. The deeper we go into the gravity the more similar-ities with the Yang-Mills theory we find with some important differences though. Fromthis perspective let us collect basic formulas of Yang-Mills theory. The main object inYang-Mills theory is the gauge potential Am that takes values in some Lie algebra, say g.We will treat it as a degree-one form A = Amdx

m with values in the adjoint representationof g. The index of the Lie algebra is implicit but we can always recover it A = AItI withtI being the generators of g, i.e. there is a Lie bracket [tI , tJ ] = fIJ

K tK.There can also be matter fields, i.e. fields taking values in arbitrary representation

of g. For example, let φ(x) = φa(x) be a vector in some vector space V that carriesa representation ρ of g, i.e. ρ : g → End(V ), which means that we have matricesρ(tI)

ab associated with each of the generators tI such that [ρ(tI), ρ(tJ )] = ρ([tI , tJ ]) =

fIJK ρ(tK), i.e. the matrix commutator is expressed via the Lie bracket and hence in

terms of the structure constants.In the table below we collect some formulae that we will use many times in what follows

13We omit the gravitational constant everywhere from our formulae. Our excuse is that we are notgoing to compute the precession of the perihelion of Mercury or anything like that in these notes.

14The integration of differential forms is explained at the end of Appendix D.

21

description formula

gauge transformation(ǫ is a zero-form with values in g, ǫ = ǫItI)

δA = Dǫ ≡ dǫ+ [A, ǫ]δφ = −ρ(ǫ)φ

curvature or field strength F (A) = dA+ 12[A,A]

covariant derivative Dφ = dφ+ ρ(A)φ

generic variation δA of F δF = DδA ≡ dδA+ [A, δA]

gauge variation of F δF = [F, ǫ]

gauge variation of Dφ δDφ = −ρ(ǫ)Dφ

Bianchi identity DF ≡ dF + [A, F ] ≡ 0

Jacobi identity [A, [A,A]] ≡ 0

the commutator of two D′s(• is a placeholder)

D2• = F•, D2φ = ρ(F )φ

For example, the Jacobi identity acquires a simpler form [A, [A,A]] ≡ 0 because A is aone-form and hence, [Am, [An, Ak]] dx

m∧dxn∧dxk implicitly imposes anti-symmetrizationover the three slots, which is the Jacobi identity. Analogously, D2 computes the commuta-tor of two D’s, DD = DmDn dx

m∧dxn ≡ 12[Dm, Dn] dx

m∧dxn, which is the field-strength.

Back to gravity. The theory of gravity in terms of vielbein/spin-connection variablesmust be invariant under the local Lorentz transformations. Now we can simply say thatωa,b is a gauge field (Yang-Mills connection) of the Lorentz algebra, so(d−1, 1). Denotingthe generators as Lab = −Lba we have the following commutation relations

[Lab, Lcd] = Ladηbc − Lbdηac − Lacηbd + Lbcηad . (3.23)

The Yang-Mills connection is then ω = 12ωa,bLab, which already looks like spin-connection.

For a moment we will treat ea as a vector matter, i.e. with ρ given by ρ(Lab)cd =

−ηadδcb + ηbdδca. As a connection, ω possesses its own gauge parameter ǫ = 1

2ǫa,bLab.

Specializing the formulas from the table above we find the gauge transformations

δωa,b = dǫa,b + ωa,c ǫ

c,b + ωb,c ǫ

a,c ≡ Dǫa,b , (3.24)

δea = −ǫa,b eb , (3.25)

which correspond to infinitesimal Lorentz rotations. The last line is exactly (3.7). Thetransformation law for the spin-connection can be derived without making any reference tothe Yang-Mills rules — one can apply the same reasonings as for the Christoffel symbols,i.e. use (3.25) and the fact that DmV

a must be a tensor quantity (Lorentz vector in theindex a).

The Yang-Mills field-strength F (ω) is exactly F a,b(ω) found above, (3.17). The torsionconstraint T a = 0, (3.16), is just the condition for the covariant derivative Dea of ea to

22

vanish. We also find that DF a,b ≡ 0 as a Bianchi identity. Taking into account therelation between F a,b and the Riemann tensor we recover the second Bianchi identityD[mRnr],ku ≡ 0.

In particular we can now solve the problem of extending Dirac Lagrangian to curvedmanifolds since the covariant derivative can act in any representation of the Lorentzalgebra. To be precise, we define fiber spinor field ψa(x), the fiber γ-matrices γa = γa

ab ,

γa, γb = 2ηab. Then the generators of the Lorentz algebra in the spinor representationare given by ρ(Lab) =

14[γa, γb] and the covariant derivative acts as

Dψa = ∂mψa +

1

2ωa,bρ(Lab)

ab ψ

b . (3.26)

Finally, the Dirac action on a curved background reads,

SD[ψ, e, ω] =

∫det e (iψγaena

−→Dnψ − iψγaena

←−Dnψ −mψψ) . (3.27)

What are the symmetries of the frame-like action (3.22)? All the fiber indices arecontracted with the invariant tensor ǫab...u of the Lorentz algebra and ea as well as F a,b(ω)transform homogeneously under local Lorentz rotations, i.e. like a vector and a rank-twoantisymmetric tensor. This implies that the action has local so(d− 1, 1)-symmetry. It isalso diffeomorphism invariant since it is an integral of a top-form.

There are still some subtleties that prevent one from simply stating that gravity isthe Yang-Mills theory. Namely, ωa,b is a function of vectorial matter ea via the torsionconstraint, (3.16); ea is a one-form rather than pure vector matter; eam must be invertiblesince det g 6= 0; the action does not have the Yang-Mills form. Nevertheless, by going tothe first-order formulation of gravity one can further improve the interpretation of gravityas a gauge theory.

Note on first-order actions. We are not aiming at rigorous definitions here. Thefield equations are usually second-order P.D.E.s for bosonic fields. We call the actionsthat immediately lead to second-order equations the second-order actions. For example,classical action for a free particle

∫12qiq

i, the Fronsdal action, (2.19), or the Einstein-Hilbert action are second-order actions because the variational equations are of the second-order.

Let us begin with the free particle. The Hamiltonian is H = 12pip

i, pi = qi. We canexpress the Lagrangian back using L = pq − H , where we would like to treat pi as anindependent variable for a moment, so we have

S(q, p) =

∫(qi − 1

2pi)pi . (3.28)

Now there are two variational equations

δS

δpi= qi − pi = 0 ,

δS

δpi= pi = 0 . (3.29)

The first equation is algebraic with respect to momenta pi and is solved as pi = qi. Thenthe second equation reduces to pi = qi = 0 as desired.

23

To make notation coherent we can use dqi, where d = dt ∂∂t

instead of qidt, so that wetreat qi as a vector valued zero-form over one-dimensional manifold, which is the world-line of the particle parameterized by t. We can also introduce a one-dimensional einbeine = dt to write

S(q, p) =

∫(dqi − 1

2epi)pi . (3.30)

This is how a typical first-order action looks like. The ideology is that one introducesadditional fields, the analogs of momenta p, such that the new, first-order, action dependson the original fields and momenta. The action now contains first-order derivatives only.The equations for momenta are algebraic and express momenta as first-order derivativesof the original fields. On substituting the solutions for the momenta into the action onegets back to the original action. In many cases the advantage of the first-order approachis that the action is simpler, less nonlinear and the new fields, momenta, as independentfields may have certain interpretation (this is what happens to ωa,b).

As an example, it is well-known that in the case of gravity one can treat Γkmn as an

independent variable in the action (Palatini formulation), writing

SP (g,Γ) =

∫ √det g gmnRmn(Γ) . (3.31)

The equations of motion for Γ are equivalent to ∇mgnk = 0 and imply that Γk

mn is theLevi-Civita connection. On substituting this to the action one gets back to the pureEinstein-Hilbert.

One can do something more by replacing gmn as independent variable with15 gmn =√det g gmn. Then the action for gravity is schematically g(∂Γ + Γ2), i.e. at most cubic.

All non-polynomial nonlinearities of gravity are removed by using the first-order approachand new appropriate variable g. There are two sources of nonlinearities. The first one, isin√det g gmn. The second one arises when solving for Γ as one has to invert the metric.

Back to gravity again. Let us take the route of first-order actions and see if wecan treat ωa,b as an independent variable (the analog of Palatini formulation in terms ofvielbein and spin-connection). Generally, one cannot just isolate a bunch of derivativesand fields (the expression for ω in terms of ea) and call it a new field to get the first-orderformulation. Fortunately, this is not the case with ω. The variation of the action

Sf (e, ω) =

∫F a,b(ω) ∧ ec ∧ ... ∧ eu ǫabc...u , (3.32)

where ω is now an independent field, reads16

δSf(e, ω) = (d− 2)

∫ (δωa,b ∧ T c ∧ ... ∧ eu + F a,b(ω) ∧ δec ∧ ... ∧ eu

)ǫabc...u , (3.33)

15This is a purely algebraic change of variable, nonlinear though. It is invertible in d > 2.16It is very useful not to split covariant derivativeD into two piecesD = d+ω. This can be supported by

the Stokes theorem 0 =∫d(Ap∧Bq) =

∫dAp∧Bq+(−)pAp∧dBq , where p+q = d, which holds true for

covariant derivatives as well, provided the integrand is a scalar. First, one checks that if I is a scalar, i.e.in a trivial representation of the Lie algebra we are considering, then dI ≡ DI. Then if I is a composite,e.g., I = Ap∧Bq , thenD satisfies the chain rule, which gets modified by a sign factor in front of the secondterm since all the objects are differential forms, i.e. d(Ap∧Bq) = D(Ap∧Bq) = DAp∧Bq+(−)pAp∧DBq .

24

where we used that δF a,b = Dδωa,b and integrated by parts to find T a = Dea. Assumingthe frame field be invertible we find the following equations

δωa,bm : T a = Dea = dea + ωa,

b ∧ eb = 0 , (3.34)

δeam : F a,bmn e

nb = 0 . (3.35)

The first equation allows us to solve for the spin-connection. Under the condition thatT a = 0 the second equation, when rewritten in the metric-like language, gives Rmn = 0,which is the vacuum Einstein equation. Up to the moment when we have to solve thetorsion constraint, (3.16), ωa,b can be treated as a Yang-Mills connection of the Lorentzalgebra.

Let us note that the first-order action is polynomial as compared to the second orderaction where the nonlinearities come from ω(e) that involves inverse of the vielbein.

It is worth stressing that second-order and first-order approaches may lead to differentresults under certain conditions. For example, if we add matter, such as spin-1

2fields,

(3.27), to the gravity action, i.e. S = Sf + SD then the torsion constraint gets modifiedinasmuch as ωa,b contributes to the matter action, SD = SD[ψ, e, ω]. In the secondorder approach ω = ω(e). In the first order approach instead of T a = 0 we find T ∼ψψ, i.e. the torsion is fixed in terms of the matter fields. One can still solve for ω =ω(e, ψ). Restoring the gravitational constant one finds the difference between the first-order and second order gravity to be quadratic in the gravitational constant which hasnever been tested experimentally. Within the second-order approach one can alwaysreproduce the corrections due to ω(e)−ω(e, ψ) 6= 0 by adding them to the action by hand.In supergravity, thinking of ω as an independent field leads to more compact expressions.Namely one can start from the second-order approach and then find that certain termshave to be added to the action to make it invariant under super-transformations. Theseterms can be reproduced automatically by considering ω as an independent field, i.e.within the first-order approach.

The cosmological term Λ√det g can be represented in the frame-like form

SΛ = Λ

∫ea ∧ eb ∧ ... ∧ eu ǫabc...u . (3.36)

In some applications the gravity Lagrangians are allowed to have a more general form,suggested first by Lovelock, [93]. One adds scalar polynomials in the Riemann tensorsuch that the equations of motion are still second-order. The Lovelock terms have thefollowing simple form within the frame-like approach

SL,n =

∫F a,b ∧ ... ∧ F c,d︸ ︷︷ ︸

n

∧ef ∧ eg ∧ ... ∧ eu ǫab...cdfg...u . (3.37)

In particular it is easy to see that there are [d/2] Lovelock terms in dimension d with SL,1

corresponding to the pure Einstein-Hilbert. The equations of motion involve at most twotime derivatives. Indeed, each F , F = dω + ..., is of the second order with respect to e,F0i = ωi + ..., i = 1...d − 1. Since the ∧-product anti-symmetrizes over the indices, F0i

can appear only once in F ∧ ... ∧ F .

25

In even dimension d = 2n the top Lovelock term is topological

SL,n =

∫F a,b ∧ ... ∧ F c,d︸ ︷︷ ︸

n

ǫab...cd , (3.38)

its variation vanishes up to boundary term because of δF = Dδω and DF ≡ 0.Other possibilities include Yang-Mills-type action

SL,n =

∫F a,bmnF

c,drk ηacηbdg

mrgnk =

∫tr(FmnFrk)g

mrgnk , (3.39)

which turns out to be higher-derivative (it is not topological or of Lovelock type) anddoes not lead to the Einstein-Hilbert action.

More on gravity as a gauge theory. That spin-connection ωa,b and vielbein ea areboth one-forms makes one expect they should have a similar interpretation. Later we findthat it will be possible to consider all one-forms as taking values in some Lie algebra. Welook for a unifying connection A = 1

2ωa,bLab + eaPa, where Pa are generators associated

with gauge field ea. We already know the commutator [L, L] and also know that ea

behaves as a vector of so(d − 1, 1), which fixes the commutator [L, P ]. There are not somany things one can write for [P, P ] and we are left with a one-parameter family

[Lab, Lcd] = Ladηbc − Lbdηac − Lacηbd + Lbcηad ,

[Lab, Pc] = Paηbc − Pbηac ,

[Pa, Pb] = −ΛLab ,

(3.40)

where Λ is some constant and the Jacobi identities are satisfied for any Λ. Freedom inrescaling the generators leaves us with three distinct cases: Λ > 0, Λ < 0 and Λ = 0.These three cases can be easily identified with de-Sitter algebra so(d, 1), anti-de Sitteralgebra so(d− 1, 2) and Poincare algebra iso(d− 1, 1), respectively.

That Λ = 0 corresponds to iso(d − 1, 1) is obvious. Let TAB = −TBA, where A,B, ...range over a and one additional direction, denoted by 5, i.e. A = a, 5, be the generatorsof so(d, 1) or so(d− 1, 2). They obey

[TAB, TCD] = TADηBC − TBDηAC − TACηBD + TBCηAD . (3.41)

Defining Lab = Tab,√|Λ|Pa = Ta5 we find (3.40) with the last relation being [Pa, Pb] =

−η55|Λ|Lab, which explains the minus.The Yang-Mills curvature17 F = 1

2Ra,bLab+T

aPa and gauge transformations δA = Dǫ,where ǫ = 1

2ǫa,bLab + ǫaPa read

Ra,b = dωa,b + ωa,c ∧ ωc,b − Λeaeb , T a = Dea , (3.42)

δωa,b = Dǫa,b − Λeaǫb + Λebǫa , δea = Dǫa − ǫa,b eb , (3.43)

17We reserve F a,b for the Riemann two-form, while Ra,b contains the cosmological term.

26

where D is the Lorentz covariant derivative d+ ω. The torsion is now one of the compo-nents of the Yang-Mills field-strength! According to the general Yang-Mills formulae, thecurvatures transform as

δRa,b = −ǫa,cRcb − ǫb,cRa,c − ΛT aǫb + ΛT bǫa , (3.44)

δT a = −ǫa,b T b +Ra,b ǫ

b . (3.45)

The Bianchi identity for the Yang-Mills field-strength, DF = 0, when written in compo-nents is

DT a − em ∧Ra,m ≡ 0 , (3.46)

DRa,b − ΛT a ∧ eb + ΛT b ∧ ea ≡ 0 (3.47)

If the torsion constraint (3.16) is imposed the first equation simplifies18 to em ∧F a,m ≡ 0,which otherwise results from differentiation of torsion (3.17). The second one simplifies toDF a,b ≡ 0, which is the second Bianchi identity for the Riemann tensor, ∇[uRmn],kr ≡ 0.Consequently, all useful relations automatically arise when both e and ω are combinedinto a single connection.

The cosmological term (3.36) is included into the action (3.32) if F is taken to be thefield-strength of the (anti)-de Sitter algebra, Ra,b, rather than Poincare one, i.e.

S =

∫Ra,b(ω) ∧ ec ∧ ... ∧ eu ǫabc...u = Sf + SΛ . (3.48)

There appeared a new gauge symmetry with parameter ǫa, the local translations, whichwe did not observe before. However, the action (3.32) is invariant under so(d− 1, 1)-partof gauge transformations, i.e. ǫa,b, and it is not invariant under local translations with ǫa

δSf = −(d− 2)(d− 3)

∫Ra,b(ω) ∧ T c ∧ ǫc ∧ ef ∧ ... ∧ eu ǫabcf...u+ (3.49)

+2Λ(d− 3)

∫T a ∧ ǫb ∧ ec ∧ ... ∧ eu ǫabc...u . (3.50)

It is not a new symmetry. Local translations become a symmetry of the action whentorsion is zero19, T a = 0. We stress that T a = 0 is not a dynamical equation, it is aconstraint that allows one to solve for ωa,b as a function of ea.

When torsion is zero the local translations can be identified with diffeomorphisms.Indeed, there is a general identity20

LξA = D(iξA) + iξF (A) , (3.51)

i.e. the Lie derivative of any Yang-Mills connection A = AItI can be represented as a sumof gauge transformation with ǫ = iξA, i.e. ǫ

I = ξmAIm, and a curvature term. Specializing

to our case we derive

Lξea = δξe

a + iξTa , Lξω

a,b = δξωa,b + iξR

a,b , (3.52)

18Mind that eb ∧Ra,b can be replaced by eb ∧ F a,b since ea ∧ ea ≡ 0.19Let us note that there is something special about d = 3. For example, for Λ = 0 we have δ

∫F a,b ∧

ec ǫabc =∫F a,b ∧Dǫc ǫabc which vanishes upon integrating by parts and using DF a,b ≡ 0.

20Remember that Lξ = diξ + iξd, where Lξ and iξ are Lie and inner derivatives, respectively, seeAppendix D. Then, one completes iξdA to F (A) which completes d(iξA) to a gauge transformation.

27

where δξ means the gauge variation with ǫa = ξmeam and ǫa,b = ξmωa,bm . When torsion

is zero we have Lξea = δξe

a, i.e. diffeomorphisms acting on ea can be represented asparticular gauge transformations. This is in accordance with the invariance of the actionunder local translations for vanishing torsion. Diffeomorphisms acting on ωa,b are notequivalent to gauge transformations because Ra,b, which is related to the Riemann tensor,is generally non-zero. This does not cause a problem since the dynamical variable is thevielbein. A diffeomorphism performed on e induces a diffeomorphism for gmn = eamηabe

bn.

That there are three ways, (3.40), to unify ωa,b and ea within one Lie algebra is directlyrelated to the fact there are three most symmetric solutions to Einstein equations withcosmological constant Λ. These are de Sitter space, Λ > 0, anti-de Sitter space, Λ < 0,and Minkowski space, Λ = 0.

Despite the unification of ωa,b and ea into a single Yang-Mills connection there isan important difference between the two. We use ωa,b to construct Lorentz-covariantderivative D and couple matter to gravity, e.g. as in (3.27), but we do not use ea insideD. The frame field is always outside and is used to built a volume form and contractindices. Let us mention that within the higher-spin theory the difference between ωa,b

and ea to some extent vanishes as we will see that ea does contribute to the covariantderivative!

Most symmetric background is equivalent to dA + 1

2[A,A] = 0. The impor-

tant observation is that de Sitter, anti-de Sitter and Minkowski space-times are solutionsof F (A) = 0 with A = 1

2ωa,bLab + eaPa being the gauge field of the corresponding sym-

metry algebra where the commutation relations are given in (3.40) and Λ distinguishesbetween the three options. In terms of the Riemann tensor these space-times are definedby the following constraint

Rmn,rk = Λ(gmrgnk − gnrgmk) . (3.53)

Within the frame-like approach this corresponds to

F (A) = 0 ⇐⇒T a = 0F a,b(ω(e)) = Λea ∧ eb , (3.54)

where ω is expressed in terms of e via the torsion constraint and the second equationimposes (3.53) once we remember the relation between F a,b and Rmn,rk. This is equivalentto F (A) = 0. As always it is implied that det eam 6= 0.

It is not hard to write down some explicit solutions. If Λ = 0, i.e. the space-time isMinkowski, a useful choice is given by Cartesian coordinates

eam dxm = δam dx

m = dxa , ωa,bm dxm = 0 , (3.55)

i.e., the vielbein is just a unit matrix and there is no difference between world and fiber.The spin-connection is identically zero. This choice leads to gmn = ηmn and Γk

mn = 0.If Λ 6= 0 a useful choice is given by the Poincare coordinates xm = (z, xi) where

gmn =1

|Λ|1

z2(dz2 + dxidxjηij) =

1

|Λ|1

z2(dxmdxnηmn) , (3.56)

28

where z is an analog of radial direction and xi are the coordinates on the leaves of constantz. Then we can use, for example,

eam dxm =

1√|Λ|

1

zδam dx

m ωa,bm dxm = −1

z

(δamη

bz − δbmηaz)dxm . (3.57)

Note that gmn = eamebnηab applies, of course.

As a side remark let us take the most general action composed of Lovelock-type terms,(3.37), which reads

S =∑

n

cnd− 2n

SL,n . (3.58)

It is interesting that it can have (anti)-de Sitter spaces with different cosmological con-stants as solutions. Indeed, assuming that the torsion is zero and taking the variationwith respect to ea we get

δS =∑

cn

∫F a,b ∧ ... ∧ F c,d ∧ δef ∧ eg ∧ ... ∧ euǫabcdfg...u . (3.59)

In looking for the (anti)-de Sitter type solutions we replace F a,b with Λea ∧ eb to find

0 =∑

cnΛn = (Λ− Λ1)...(Λ− Λn) . (3.60)

There are further improvements possible when cosmological constant is non-zero, theMacDowell-Mansouri-Stelle-West approach, which is reviewed in extra Section 12.2.

Summary. We have shown that the tetrad ea and spin-connection ωa,b can be unified asgauge fields of Poincare or (anti)-de Sitter algebra, A = 1

2ωa,bLab + eaPa. The Yang-Mills

curvature then delivers constituents of various actions and contains Riemann two-formand torsion.

There are at least three ways to treat Yang-Mills connections:

• As in the genuine Yang-Mills theory, i.e.∫tr(FmnF

mn) + matter.

• As in Chern-Simons theory. We are in 3d with the action∫tr(FA− 1

3A3).

• As in gravity. Here we found several options how to treat vielbein and spin-connection. The least we can do is to say that ωa,b is an so(d− 1, 1)-connection andea is a vector-valued one-form. Another option is to unify ωa,b and ea as gauge fieldsof one of the most symmetric Einstein’s vacua, i.e. (anti)-de Sitter or Minkowski.The case of (anti)-de Sitter is less degenerate since the symmetry algebras are semi-simple. Any theory can be re-expanded over one of its vacuum and the expansion iscovariant with respect to the symmetries of the vacuum. In the case of Einstein the-ory, depending on the cosmological constant, there are three maximally symmetricvacua.

We found several reasons to replace metric with the vielbein or frame and spin-connection.

29

1. To have freedom in introducing general basis in the tangent space.

2. To make transition from a generic curved coordinates to the ones of the Einstein’selevators, where the local physics is as in SR. This leads us to the idea that GRis a localized (gauged) version of SR. At any rate we expect that we should gaugethe Lorentz algebra so(d− 1, 1). This makes us feel that GR should be close to theYang-Mills theory. There are also important differences with the Yang-Mills theory,which we discuss below.

3. To make half-integer spin fields, in particular matter fields, interact with gravity.This is the strongest motivation, of course. Since it is the frame-like approach thatallows matter fields to interact with gravity it is promising to stick to this approachand look for its generalization to fields of all spins.

The similarities and distinctions between gravity and Yang-Mills theory include

+ Spin-connection is a gauge field of the Lorentz algebra.

− On-shell it is not an independent propagating field, rather it is expressed in termsof the vielbein field via the torsion constraint T a = 0.

+ The matter fields interact with ω through the covariant derivative, i.e. minimally,e.g. Dψ, like in Yang-Mills theory.

+ The action is cooked up from the Yang-Mills curvatures.

− In any case the action does not have the Yang-Mills form.

− There is a condition det eam 6= 0, i.e. det gmn 6= 0, that is hard to interpret withinthe Yang-Mills theory.

+ One can unify both vielbein and spin-connection as gauge fields of some Lie algebra.

− There are several options to achieve that (Poincare, de Sitter or anti-de Sitter).

− While ωa,b appears in covariant derivative D only, vielbein ea is always around whenbuilding a volume form or contracting indices, so the unification of e and ω withinone gauge field is not perfect — their appearance is different, both in the gravityand in the matter Lagrangians.

− The local translation symmetry associated with frame field ea becomes a symmetryof the action only when the torsion constraint is imposed, then it can be identifiedwith diffeomorphisms.

− The full group structure for a diffeomorphism invariant theory with some internallocal symmetry (Yang-Mills or gravity in terms of ea and ωa,b) is that of a semidirectproduct of diffeomorphisms and local symmetry group, see extra Section 12.3.

30

4 Unfolding gravity

Let us abandon the action principle and concentrate on the equations of motion. Theappropriate variables we need are vielbein ea and so(d − 1, 1) gauge field ωa,b, spin-connection. As we have already seen, they can be viewed as the gauge fields associatedwith either Poincare or (anti)-de Sitter algebra, the gravity then shares many featuresof Yang-Mills theory. We aim to write the Einstein equations by making use of thelanguage of differential forms. Particularly, it means that field equations are necessarilyof first order. It may seem to be too restrictive since many known dynamical equationsof interest contain higher derivatives. However, pretty much as any system of differentialequations can be reduced to a first order form by means of extra variables, so is anyclassical field theory can be rewritten in differential form language by virtue of auxiliaryfields. In practice one typically needs infinitely many of those. Such equations are calledthe unfolded, [40, 41], and it is in this form that the Vasiliev theory is given in. Theformulation of gravity obtained in this section admits a natural extension to all higher-spin fields.

Our starting point is the torsion constraint, (3.16), and the definition of the so(d−1, 1)-curvature, (3.17),

T a = Dea = dea + ωa,b ∧ eb = 0 , (4.1)

F ab = dωa,b + ωa,c ∧ ωc,b . (4.2)

The Einstein equations without matter and the cosmological constant

Rmn −1

2gmnR = 0 ⇐⇒ Rmn = 0 (4.3)

are equivalent in d > 2 to Rmn = 0 and hence

Rmn = 0 ⇐⇒ F a,bmn(e

−1)nb = 0 . (4.4)

There are two clumsy properties of the latter expression: (i) we had to undress thedifferential form indices of the curvature two-form; (ii) we needed the inverse of eam tocontract indices, i.e. from the Yang-Mills point of view we had to take the ’inverse ofthe Yang-Mills field’. One may ask a naive question: whether is it possible to formulatethe gravity entirely in the language of differential forms and connections? Indeed, this ispossible and it is the starting point for the higher-spin generalizations. In Section 6 weexplain that the equations formulated solely in the language of differential forms, unfoldedequations, have a deep algebraic meaning. For a moment let us just explore this pathblindly.

First of all, the Riemann tensor Rmn,kr is traceful and has the following decomposition

Rmn,kr = Wmn,kr + α(gmkRnr − gnkRmr − gmrRnk + gnrRmk) + β(gmkgnr − gnkgmr)R ,

α =2

d− 2, β = − 1

(d− 2)(d− 1), (4.5)

Rmn,krgnr = Rmk , Rmkg

mk = R , (4.6)

31

where Rmk is the Ricci tensor, R is the scalar curvature and Wmn,kr is the traceless partof the Riemann tensor, called Weyl tensor. The coefficients are fixed by the normalizationin the last line. Weyl tensor has the same symmetry properties as the Riemann one, i.e.

Wmn,kr = −Wnm,kr = −Wmn,rk , W[mn,k]r ≡ 0 , (4.7)

where the second property in (4.7) is the algebraic Bianchi identity. The Weyl tensor isby definition traceless

Wmn,krgnr ≡ 0 . (4.8)

In the Young diagram language21 the Weyl tensor is depicted as the ’window’-like diagram

Wmn,kr : Rnr : ⊕ • R : • (4.9)

The free of matter Einstein equations imply Rmn = 0, but the whole Riemann tensor,of course, may not be zero. Vanishing Riemann tensor, Rmn,kr = 0, describes emptyMinkowski space. While Rmn = 0 has a rich set of solutions corresponding to variousconfigurations of the gravitational field, e.g. gravitational waves, black holes etc. Thedifference between very strong Rmn,kr = 0 and Rmn = 0 is exactly the Weyl tensor. Onecan say that it is the Weyl tensor that is responsible for the richness of gravity. Thecrucial step is the equivalence of the two equations

Rmn = 0 ⇐⇒ Rmn,kr =Wmn,kr , (4.10)

where Wmn,kr has algebraic properties of the Weyl tensor otherwise left unspecified at apoint. While in the first form the equation directly imposes Rmn = 0, in the second formit tells us that the only non-zero components of the Riemann tensor are allowed to bealong the Weyl tensor direction, i.e. the Ricci, Rmn must vanish. Formally, taking traceof the second equation one finds Rmn,krg

nr = Rmk =Wmn,krgnr = 0. It is also important

that the second Bianchi identity, ∇[uRmn],kr ≡ 0, implies

∇[uWmn],kr ≡ 0 , (4.11)

i.e. Wmn,kr is arbitrary at a point but its first-derivatives are constrained. Formally, (4.11)can be viewed as one more consequence of (4.10).

The general idea behind the ’unfolding’ of gravity is that instead of specifying whichcomponents of the Riemann tensor and its derivatives have to vanish we can parameterizethose that do not vanish by new fields. This is applicable to any set of fields subject tosome differential equations. Instead of imposing equations directly we can specify whichderivatives of the fields may not vanish on-shell.

Let us transfer the above consideration to the frame-like approach. Instead of Rmn,kr

we have two-form F ab(ω). Converting the differential form indices of the F ab ≡ F abmn dx

m∧dxn to the fiber we get a four-index object

F ab|cd = −F ba|cd = −F ab|dc = F abmne

mcend . (4.12)

21An introductory course on the Young diagrams language is in the Appendix E. It is really necessaryto have some understanding of what the possible symmetry types of tensors are in order to proceed tothe higher-spin case.

32

When no torsion constraint is imposed, F ab|cd has more components than the Riemanntensor. It is antisymmetric in each pair and there is no algebraic Bianchi identity implied.We find the following decomposition into irreducible components

F ab|cd ∼ ⊗ = ⊕(

⊕)⊕(

⊕ ⊕ •)

(4.13)

When torsion constraint is imposed one derives the following consequence, the algebraicBianchi identity, (3.17), (3.20), which we will refer to as the integrability constraint

0 ≡ ddea = −d(ωab ∧ eb) =⇒ F a,b ∧ eb ≡ 0 . (4.14)

The components with the symmetry of the first three diagrams do not pass the integra-bility test as they are too antisymmetric. For example, if F ab = em ∧ enCa,b,m,n, whereCa,b,m,n is antisymmetric then F ab∧ eb = em∧ en ∧ eb Ca,b,m,n, which does not vanish sinceeam is invertible. Analogously22, if F ab = em ∧ enCam,n,b, where Caa,b,c has the symmetryof the second component and contains the trace, which is the third component, we findF ab ∧ eb = em ∧ en ∧ ebCam,n,b, which does not vanish identically. Since F ab is related tothe Riemann tensor when the torsion constraint is imposed, it comes as no surprise thatthe last three components form a decomposition of the Riemann tensor into irreducibletensors. From now on the torsion constraint is implied. Analogously to (4.5), the (fiber)Riemann tensor F ab|cd can be decomposed as

F ab|cd = Cab,cd + α(ηacRbd − ηbcRad − ηadRbc + ηbdRac) + 2β(ηacηbd − ηbcηad)R ,(4.15)

F am|bm = Rab , Rn

n = R , (4.16)

Cab,cd : , Rab : ⊕ • , R : • (4.17)

We treat Cab,cd, Rab, R as zero-forms. Cab,cd, which is the fiber Weyl tensor, has thesymmetry of the Riemann tensor, i.e. of the window Young diagram, but it is traceless,Cam,b

m ≡ 0. The trace of the fiber Riemann tensor is the fiber Ricci tensor Rab, whosetrace is the scalar curvature R, the latter coincides with R because it is a scalar.

It is easy to see that (4.15) can be rewritten solely in terms of differential forms as

F ab = em ∧ enCab,mn + αe[a ∧ enRb],n + 2βea ∧ ebR (4.18)

and it does not restrict F ab, just telling which components of F ab may in principle benon-zero when the torsion constraint is imposed.

22It is obvious that eamVa = 0 implies Va = 0 since the vielbein is invertible, e.g. we can always chooseeam = δam at a point. Similarly, operators of the form em ∧ ... ∧ enC

...m...n just antisymmetrize overthe indices m, ..., n that are contracted with vielbein one-forms. Decomposing C...m...n into irreduciblesymmetry types one checks using the Young properties which of the components are annihilated. Say,for Ca|b equation em ∧ enCm|n = 0 implies that the antisymmetric component of Ca|b must vanish whilethe symmetric one is arbitrary.

33

The Einstein equations are equivalent to Rab = 0. The same reasoning as in (4.10),compared to (4.4) forces us to require that the only non-zero components of F ab shouldbe given by the Weyl tensor. Namely, we just remove Rab, R from the r.h.s. of (4.18)

F abmn = emmennC

ab,mn ⇐⇒ F ab = em ∧ enCab,mn , (4.19)

which is equivalent to (4.4) in the same way as it was the case for (4.10).Using the ambiguity in presentation of mixed-symmetry tensors23 instead of Cab,cd,

which is manifestly antisymmetric in pairs of indices we can switch to C ′ac,bd, which ismanifestly symmetric in pairs,

Cab,cd = −Cba,cd = −Cab,dcab dc

C ′ac,bd = C ′ca,bd = C ′ac,db . (4.20)

In what follows we will use the symmetric basis for tensors, which is more convenient forhigher-spin fields, and we write Caa,bb instead of C ′aa,bb. In the symmetric basis24 (4.19)reads

F ab = em ∧ enCam,bn . (4.21)

The r.h.s. of (4.21) manifestly satisfies the algebraic Bianchi identity (4.14), but it is notso for the differential Bianchi identity, an analog of (4.11). The second Bianchi identityis a consequence of DF ab ≡ 0, which implies (recall that Dea = 0)

0 ≡ DF ab = D(em ∧ enCam,bn) = em ∧ en ∧DCam,bn . (4.22)

We can either stop here and supplement (4.21) with (4.22) or try to analyze this in-tegrability condition. (4.22) is a restriction on the derivatives of the Weyl tensor. WereCab,cd arbitrary we would find several tensor types to appear in DmC

ab,cd. This is equiv-alent to analyzing the second term of the Taylor expansion of Caa,bb(x) and decomposingTaylor coefficients into irreducible so(d − 1, 1) tensors. In doing so it is convenient totransfer the world index of Dm to the fiber and define

Baa,bb|c = emcDmCaa,bb . (4.23)

We use the slash notation within a group of tensor indices (e.g., Baa,bb|c) to split the indices(tensor product) into the groups of indices in which the tensor is irreducible. Since wedropped the second Bianchi identity for a moment there are no relations between ab, cdand m, i.e. as the tensor of the Lorentz algebra it has the following decomposition intoirreducibles

Baa,bb|c ∼ ⊗ = ⊕(

⊕)

(4.24)

23This is a special feature about mixed-symmetry tensors that allows us to use dual Young tableauxpresentation. More detail on mixed-symmetry tensors can be found in Appendix E.

24A typical exercise on Young symmetry is to show that the r.h.s. is antisymmetric in ab, which isnot manifest. The expression is antisymmetric if its symmetrization vanishes. Symmetrizing over ab weget Cam,bn + Cbm,an which is almost the Young condition Cam,bn + Cbm,an + Cab,mn ≡ 0. Hence we get0 ≡ F ab + F ba = −em ∧ enC

ab,mn. Remembering that em ∧ en is antisymmetric in mn while Cab,mn issymmetric we get the desired em ∧ enC

ab,mn ≡ 0.

34

To parameterize all three components we can introduce three zero-forms Caaa,bb, Baa,bb,c

and Baa,b

DCaa,bb = em

(Caam,bb +

1

2Caab,bm

)+ emB

aa,bb,m+

+ ebCaa,b + eaCbb,a − 2

d− 2em

(ηaaCbb,m + ηbbCaa,m − 1

2ηabCab,m

). (4.25)

The first two terms project25 onto , so do the first two terms in the second line aswell. The resulting tensor is not traceless, the trace projector is imposed with the helpof the last group of terms. The projector for emB

aa,bb,m is trivial. We see that while it iseasy to say what Young diagrams of so(d− 1, 1)-irreducible tensors that Baa,bb|c contain,this amounts to computing the tensor product. It is much more complicated to handlethis decomposition in the language of tensors due to Young- and trace-projectors thatare generally there. Luckily, many statements can be proven in terms of Young diagramswithout appealing to the tensor language.

Back to (4.24), it is easy to see that the presence of the last two components is notconsistent with (4.22), which is equivalent to em ∧ en ∧ ecBam,bn,c ≡ 0. So we have to keepthe first term only in order to write solution to the differential Bianchi identity

DCaa,bb = em

(Caam,bb +

1

2Caab,bm

), Caaa,bb : (4.26)

Caaa,bb is the first ’descendant’ of the Weyl tensor that allows us to solve the differentialconstraint in a constructive way. The second term together with 1

2in front of it ensures

that the r.h.s. has the right Young symmetry. Again, we can either stop here or checkif there are differential constraints for Caaa,bb. So far all the equations were exact in thesense that we had not neglected any terms. In continuing the process we find a technicalcomplication. Indeed, D2 = F ∼ eeC, (4.21), so checking the integrability of the equationlast obtained we get stuck with

DDCaa,bb = em ∧ enCam,nc C

ac,bb + em ∧ enCbm,ncC

aa,cb = −em(DCaam,bb +

1

2DCaab,bm

)

That D2 ∼ eeC makes the l.h.s. nonlinear, so we have to solve eeCC = eDC1 where C1

denotes Caaa,bb. Hence we have to introduce quadratic terms on the r.h.s. of DCaaa,bb.This is what should be expected since gravity is a nonlinear theory.

25One more feature of mixed-symmetry tensors is that the operation of adding one index (taking tensorproduct) or removing (contracting with an external object) one index does not preserve the symmetryproperties. Indeed, the tensor product contains in general several irreducible components and we are freeto keep them all or to project onto one of them. For example, given T aa,b of symmetry type, we cancontract it with vielbein to get emT am,b. The resulting tensor is neither symmetric nor antisymmetric in ab

and contains both and symmetry types. It can be projected onto simply as emT am,b+emT bm,a =−emT ab,m and onto as emT am,b − emT bm,a. The operation of adding one index and projecting ontoirreducible component is more complicated. For example, ebT aa,b neither has any definite symmetry typenor is it traceless. Typically a number of terms is needed to project a contraction/tensor product of twoirreducible tensor objects onto an irreducible component. If we need an irreducible tensor of orthogonalalgebra, rather than gl(d), we have in addition to project out its traces.

35

We will use the example of gravity in the frame-like formalism as a starting pointfor higher-spin generalization. The extension to higher-spin fields is to be done at thefree level first and then we review the Vasiliev solution to the nonlinear problem. At theinteraction level fields of all spins interact with each other and no truncations of the fullsystem to a finite subset of fields is possible. Given that we will continue the gravity partof the story at the linearized level only, i.e. we will neglect D2, assuming that D2 ∼ 0.

Repeating the derivation of the constraint on the first derivatives of Caaa,bb from (4.26)in the approximation D2 = 0 we now find

DDCaa,bb = 0 = −em(DCaam,bb +

1

2DCaab,bm

). (4.27)

It can be solved analogously to the way we did before by considering all possible r.h.s. ofDCaaa,bb resulting in

DCaaa,bb = em

(Caaam,bb +

1

3Caaab,bm

)+O(eC2) aaaa, bb = (4.28)

Continuing this process we can derive one by one the following set of equations/constraintsthat describes Einstein’s gravity without the cosmological constant and matter withhigher-order corrections due to D2 6= 0 neglected

T a = Dea = 0δea = Dǫa − ǫa,b eb (4.29)

F a,b = dωa,b + ωa,

c ∧ ωc,b = em ∧ enCam,bn

δωa,b = Dǫa,b + (ǫmen − ǫnem)Cam,bn (4.30)

DCaa,bb = em

(Caam,bb +

1

2Caab,bm

)aa, bb =

(4.31)

DCa(k+2),bb = em

(Ca(k+2)m,bb +

1

k + 2Ca(k+2)b,bm

)+O(eC2) a(k + 2), bb = k

δCa(k+2),bb = −ǫa,mCa(k+1)m,bb − ǫb,mCa(k+2),bm

+ ǫm

(Ca(k+2)m,bb +

1

k + 2Ca(k+2)b,bm

)+O(ǫC2) (4.32)

where we marked those equations that are not exact with +O(eC2) label. Notice that allCa(k+2),bb transform under local Lorentz transformations.

It is worth stressing that zero-forms, like the Weyl tensor or matter fields of Yang-Mills, do not have their own gauge parameters. Nevertheless, they take advantage ofgauge fields’ parameters, but the gauge transformations do not contain derivatives.

Notice the local translation symmetry Dǫa in (4.29), (4.30), (4.32). It gets restoredsince the torsion constraint is imposed and we can interpret local translations as dif-feomorphisms. Hence, we automatically gauge the Poincare algebra when consideringequations of motion, while at the level of the action principle we had certain problems ininterpreting it as resulting from gauging of the Poincare algebra.

36

Equivalently, we can consider the first order expansion of gravity over the Minkowskibackground. The background Minkowski space can be defined by vielbein ham and by

spin-connection a,bm = −b,a

m obeying the torsion constraint and zero-curvature

T a = dha +a,b ∧ hb = 0 , da,

b +a,c ∧c,

b = 0 , (4.33)

the latter implies that the whole Riemann tensor vanishes. The convenient choice in thecase of Minkowski is given by Cartesian coordinates, where

gmn = ηmn , Γrmn = 0 , ham = δam , a,b

m = 0 . (4.34)

It is useful to define the background Lorentz-covariant derivative D = d + , whichwe denote by the same letter as the full Lorentz-covariant derivative above. That (4.33)implies D2 = 0 supports this notation as we were going to neglect D2 anyway.

The linearization of (4.29)-(4.32) over Minkowski background readsT a = Dea − hm ∧ ωa,m = 0δea = Dǫa − ǫa,b eb (4.35)

F a,b = Dωa,b = hm ∧ hnCam,bn

δωa,b = Dǫa,baa, bb = (4.36)

DCaa,bb = hm

(Caam,bb +

1

2Caab,bm

)aaa, bb =

DCaaa,bb = hm

(Caaam,bb +

1

3Caaab,bm

)aaaa, bb =

DCa(k+2),bb = hm

(Ca(k+2)m,bb +

1

k + 2Ca(k+2)b,bm

)a(k + 2), bb = k

δCa(k+2),bb = 0 (4.37)

Let us make few comments on this system. D is defined with respect to the background,d + . The linearized torsion constraint has a slightly different form because ωa,

m ∧ emyields, when linearized, two types of terms, e and ωh, the first being hidden inside thebackground Lorentz derivative, D. Analogously, when linearized, F ab has lost ωω pieceand is simply Dωa,b.

Minkowski background implies D2 = 0, so the linearized curvature Dωa,b is gauge in-variant. Therefore, Caa,bb on the r.h.s. of (4.36) should not transform under ǫa,b anymore,the same being true for all Ca(k+2),bb. This is due to the fact that D has lost dynamicalspin-connection ωa,b, of which ǫa,b is a gauge parameter. No mixing of the form ωC impliesthere is no need to rotate Ca(k+2),bb anymore and similarly is for the ǫa-symmetry. Thisall follows from linearization, of course.

The linearized Einstein equations, i.e. the Fronsdal equations for s = 2, are imposed by(4.35)-(4.36), which we will show for the spin-s generalization later. The rest of equations(and vanishing of the torsion) are constraints in a sense that they do not impose anydifferential equations merely expressing one field in terms of derivatives of the other.

Going to nonlinear level we find O(eC2) and higher order corrections on the r.h.s.of equations due to D2 ∼ eeC. Analogous type of corrections we find in the equationsdescribing higher-spin fields.

37

5 Unfolding, spin by spin

In this section we begin to move towards the linearized Vasiliev equations that describean infinite multiplet of free HS fields in AdS. Some preliminary comments are below.

When fields of all spins are combined together into the multiplet of a higher-spinalgebra the equations they satisfy turn out to be much transparent and revealing thanthe equations for individual fields. We will not follow this idea in this section ratherconsider those fields individually, spin by spin. Moreover the technique that happened tobe extremely efficient for HS fields may look superfluous for some simple cases like a freescalar field, which we consider at the end.

The idea we blindly follow in this section is to look for the frame-like (tetrad-like,vielbein-like, ...) formulation for fields of all spins. The simple guiding principle is toexpress everything in the language of differential forms. All fields in question are dif-ferential forms that may take values in some linear spaces that are viewed as the fiberover the space-time manifold. The equations of motion are required to have the followingschematic form

d(field) = exterior products of the fields themselves (5.1)

This is what we have already achieved for the case of gravity. Equations of this form arecalled the unfolded equations, [40, 41]. It is this simple idea that could have been usedto discover the frame-like formulation of gravity and yet it also leads to the nonlinearframe-like formulation of higher-spin fields. The detailed and abstract discussion is left toSection 6, where we show that such equations are intimately connected with the theoryof Lie algebras.

The structure on the base manifold that we will need is quite poor — only differentialforms are allowed to be used along with the operations preserving the class of differentialforms, i.e. the exterior derivative d and the exterior product, ∧.

The spin-two case corresponds to the gravity itself. At the nonlinear level we find aperfect democracy among fields of all spins. This not quite so at the free level because allfields propagate over Minkowski (this section) or (anti)-de Sitter space (Section 9) which isthe vacuum value of the spin-two. We have the background non-propagating gravitationalfields defined by ha, a,b, which obey (3.54) with Λ = 0, i.e. the Yang-Mills field strengthof the Poincare algebra is zero. These fields are considered to be the zero order vacuumones within the perturbation theory, while propagating fields of all spins, including thespin-two itself, are of order one, so that the equations are linear in perturbations. If wehad a master field, say W , whose components correspond to fields of all spins and the fulltheory in terms ofW , then we could say that we expanded it asW0+gW1+higher orders,where g is a small coupling constant and

W0 = 0︸︷︷︸s=0

, 0︸︷︷︸s=1

, ha, a,b, Caa,bb = 0, ..., Ca(k+2),bb = 0, ...︸ ︷︷ ︸s=2

, 0︸︷︷︸s=3

, 0︸︷︷︸s=4

, ... , (5.2)

where the sector of spin-two is non-degenerate and contains ha, a,b that obey Dha = 0and D2 = 0, D = d+, which is equivalent to having Minkowski space.

We would like to read off the part of the theory that is linear in W1 and determine W1

itself. Recall that in the case of linearized gravity W1 was found to contain one-forms ea,

38

ωa,b and infinitely many zero-forms Ca(k+2),bb that start from the Weyl tensor Caa,bb fork = 0. For the Minkowski vacuum all Ca(k+2),bb vanish. If we wish to expand the theoryover the space with say a black hole inside, then the Weyl tensor would be nonvanishing.

Each of the considered below cases consists of two parts: a quasi-derivation explainingwhy the solution has its particular form and a part with the results, where the system ofequations is written down. The relevant original references include [39, 88, 94, 95].

5.1 s = 2 retrospectively

By the example of gravity we would like to show the main steps of how one could havediscovered the frame-like gravity from the Fronsdal theory for s = 2 using the idea thatthe theory should be formulated in terms of differential forms and bearing in mind theYang-Mills theory. The starting point is a symmetric traceful field φaa, the Fronsdal fieldat s = 2, that has a gauge symmetry, (2.5),

δφaa = ∂aǫa (5.3)

and the Fronsdal equations, (2.4), specialized to s = 2. We would like to replace φaa witha yet unknown differential form e∗ of a certain degree q taking values in some tensor rep-resentation of the Lorentz algebra, denoted by placeholder ∗. The gauge transformationsare then δe∗ = dǫ∗, where ǫ∗ is a differential form of degree q − 1 that takes values in thesame Lorentz representation ∗. Comparing

δφaa = ∂aǫa ⇐⇒ δe∗ = dǫ∗ = dxm∂mǫ∗ (5.4)

we see that one index a carried by ∂a should turn into ∂m, then the leftover index shouldbelong to ǫ∗, i.e. ∗ = a = . Since e∗ must carry the same indices as its gauge parameterwe have eam and hence δeam = ∂mǫ

a. That the gauge parameter is a Lorentz vectorimmediately tells us that the frame field has a vector index too. Since the world indices ofdifferential forms are disentangled from fiber indices, to write ∂mǫ

a+∂aǫm is meaningless.We also see that the frame field must be a one-form because the gauge parameter isnaturally a zero-form. Consequently, we found

δea = dǫa . (5.5)

That world and fiber indices in eam are disentangled implies that there are no symme-try/trace conditions between a and m in eam. In particular eam contains more componentsas compared to original φaa, which is symmetric. In Minkowski space in Cartesian coor-dinates there is no difference between world and fiber indices, but formally we can usethe background hma to convert the world indices. With the help of hma we see that inaddition to the totally symmetric component to be identified with φaa the frame field,ea|b = eamh

mb, contains an antisymmetric component too, i.e.

ea|b ∼ ⊗ = ⊕ ( ⊕ •) . (5.6)

The symmetric part of the frame field transforms as the Fronsdal field. Indeed,

δea|b = ∂bǫa =⇒ δ(ea|b + eb|a) = ∂bǫa + ∂aǫb . (5.7)

39

The antisymmetric component can be a propagating field unless we manage to get rid ofit. The simplest solution is to introduce a new gauge symmetry, which acts algebraicallyand whose purpose is to gauge away the antisymmetric component completely,

δeam = ∂mǫa − ǫa,m , (5.8)

where ǫa,b = −ǫb,a and we do not care about the difference between world and fiber indicessince we can always set ham = δam at a point. To make the last expression meaningful wecan cure it as

dxm δeam = dxm ∂mǫa − dxm hbmǫa,b ⇐⇒ δea = dǫa − hb ∧ ǫa,b . (5.9)

Now we can gauge away the antisymmetric part of the frame field

δ(ea|b − eb|a) = ∂bǫa − ∂aǫb − 2ǫa,b . (5.10)

Indeed, the gauge symmetry with ǫa,b is algebraic and obviously ǫa,b has the same numberof components. Therefore, we can always impose (ea|b − eb|a) = 0 and the condition forthe left-over gauge symmetry

0 = δ(ea|b − eb|a) = ∂bǫa − ∂aǫb − 2ǫa,b (5.11)

expresses 2ǫa,b = ∂bǫa − ∂aǫb and does not restrict ǫa. So, when = 0 gauge is imposedthe whole content of the frame field is given by the Fronsdal field with its correct gaugetransformation law.

In practice we do not need to impose = 0 gauge or to go to the component form andconvert indices with the inverse background frame field hmb. What we need is a guaranteethat the theory can be effectively reduced to the Fronsdal one (at least at the linearizedlevel) and that there are no extra propagating degrees of freedom.

We used the Fronsdal theory as a starting point for the frame-like generalization,but all the statements, e.g. that the equations to be derived below do describe a spin-srepresentation of the Poincare algebra, can be made without any reference to the Fronsdaltheory. Since the frame field is needed anyway, e.g., to make fermions interact withgravity, there is no reason to go back to the Fronsdal theory once the frame-like higher-spin generalization is worked out.

Since the background may not be given in Cartesian coordinates the form of the gaugetransformations valid in any coordinate system reads

δea = Dǫa − hb ∧ ǫa,b , (5.12)

where D = d+ is the background Lorentz derivative and we recall that D2 = 0.Now we can recognize (5.12) as the combination of local translations and local Lorentz

transformations. The local translation symmetry has its roots in the Fronsdal’s symme-try, while the purpose for local Lorentz transformations is to compensate the redundantcomponents resided inside the frame field.

We found a reason for gauge parameter ǫa,b. In the realm of the Yang-Mills theory,there are no homeless gauge parameters. Hence, there must be a gauge field ωa,b, whichis a one-form and its gauge transformation law starts as δωa,b = dǫa,b + ... or equivalently

δωa,b = Dǫa,b + ... . (5.13)

40

The existence of the spin-connection goes hand in hand with the existence of ǫa,b.Given gauge transformations (5.12)-(5.13) it is easy to guess the gauge invariant field-

strengths to be

T a = Dea − hb ∧ ωa,b , F a,b = Dωa,b + ... , (5.14)

which is the torsion and the linearized Riemann two-form. The question of whether weshould impose T a = 0 and what we should write instead of ... in F a,b is dynamical. Onecan see that setting T a to zero expresses ω in terms of e and imposes no differentialequations on the latter. At this point one can repeat the analysis of the previous sectionto find that F a,b = hm ∧ hnCam,bn imposes the Fronsdal equations, etc.

The case of s = 2 is deceptive since ham is already a vielbein. The illustration with spin-two is not self contained because the background space must have been already definedin terms of ha and a,b. We can imagine that we know how to define the backgroundgeometry but we are unaware of how to put propagating fields on top of this geometry.Later we will find out that propagating field ea can be naturally combined with thebackground ha into the full vielbein.

5.2 s ≥ 2

We would like to systematically look for an analog of the frame-like formalism for thefields of any spin s ≥ 2. The starting point is the gauge transformation law and thealgebraic constraints on the Fronsdal field and its gauge parameter

δφa(s) = ∂aǫa(s−1) , φa(s−4)mnnm ≡ 0 , ǫa(s−3)n

n ≡ 0 . (5.15)

The Fronsdal field has to be embedded into a certain generalized frame field ea...q with theform degree and the range of fiber indices to be yet determined. We understand alreadythat writing dea...q = ... implies gauge transformation of the form δea...q = dξa...q−1 with ξa...q−1

being the form of degree q − 1 valued in the same representation of the fiber Lorentzalgebra. This gauge symmetry is in general reducible δξa...q−1 = dχa...

q−2 unless q = 1. Weknow that there are no reducible gauge symmetries in the case of totally-symmetric fields.Therefore, q = 1 and gauge parameter is a zero-form, ξa..., i.e. it has no differential formindices. In order to match

dxm ∂mξa... with ∂aǫa(s−1) (5.16)

the gauge parameter must be ξa(s−1), i.e. symmetric and traceless in the fiber indices. So,the frame field must be one-form e

a(s−1)1 while the gauge transformation law now reads

δea(s−1)1 = dξa(s−1) + ... , ea(s−3)m

m = ξa(s−3)mm ≡ 0 . (5.17)

Let us convert the world indices in the last formula to the fiber

δea(s−1)|b = ∂bξa(s−1) , ea(s−1)|b = ea(s−1)m hmb . (5.18)

41

As in the spin-two case the frame field contains more components than the original metric-like, Fronsdal, field. The irreducible content of ea(s−1)|b is given by26

s− 1 ⊗ = ( s ⊕ s− 2 )⊕ s− 1 , (5.19)

where the first two components, when put together, are exactly the content of the Fronsdaltensor since a double-traceless rank-s tensor φa(s) is equivalent to two traceless tensorsψa(s) and ψa(s−2) of ranks s and s− 2

φa(s) = ψa(s) +1

d+ 2s− 4ηaaψa(s−2) , φa(s−2)m

m = ψa(s−2) , φa(s−4)mnnm ≡ 0 , (5.20)

where the coefficient is fixed by the relation in the middle. In terms of field componentsthe decomposition of ea(s−1)|b into a Fronsdal-like field and the leftover traceless tensorψa(s−1),b with the symmetry of s−1 reads

ea(s−1)|b =1

sφa(s−1)b + ψa(s−1),b+

+1

s(d+ s− 4)

((s− 2)

2ηabφa(s−2)m

m − ηaaφba(s−3)mm

). (5.21)

The overall normalization is fixed in such a way that

ea(s−1)|a = φa(s) . (5.22)

(remember that ψa(s−1),a ≡ 0 and it drops out). Note the group of terms in the second lineof (5.21) which are absent in the spin-two case. The coefficients are fixed from (5.22) andfrom the tracelessness of ea(s−1)|b in a indices. In the language of differential forms withthe Fronsdal field φa(s) and ψa(s−1),b treated as zero-forms the embedding of the Fronsdalfield reads

ea(s−1)1 =

1

shm φ

a(s−1)m + hbψa(s−1),b+ (5.23)

+1

s(d+ s− 4)

((s− 2)

2haφa(s−2)c

c − ηaahmφa(s−3)mcc

). (5.24)

We see that such an unnatural within the metric-like approach condition as vanishingof the second trace

gmmgnnφmmnnr(s−4) ≡ 0 (5.25)

comes from a very natural condition within the frame-like approach — the frame fieldis an irreducible fiber tensor. The condition of the type (5.25) is a source of problemsin the interacting theory as it has to be either preserved or deformed when gmn is notMinkowski or (anti)-de Sitter but a dynamical field. In contrast, ηbbe

bba(s−3) ≡ 0, (5.17),causes no problem since ηaa is a non-dynamical object, the dynamical gravity is describedby the frame field ea1. The identification of the Fronsdal tensor as the totally symmetriccomponent of the frame field we discussed is essentially linear.

26We refer to the Appendix E.2, where the tensor product rules are discussed.

42

Combining (5.18) with (5.22) we recover the Fronsdal gauge transformation law forthe totally symmetric component of the frame field.

As in the case of spin-two, the frame field contains an additional component ψa(s−1),b.To prevent it from becoming a propagating field we can introduce an algebraic gaugesymmetry with a zero-form parameter ξa(s−1),b, so as to make it possible to gauge awayψa(s−1),b. As a tensor ξa(s−1),b is irreducible, i.e. Young and traceless, since the unwantedcomponent is like that. This correction can be written as follows

δea(s−1)1 = −hmξa(s−1),m (5.26)

and coincides for s = 2 with the action of local Lorentz rotations. Once we had tointroduce ξa(s−1),b there must be an associated gauge field ω

a(s−1),b1 , the generalization of

the spin-connection. This allows us to write the first equation immediately as27

dea(s−1)1 = hm ∧ ωa(s−1),m

1 (5.27)

together with the gauge transformations

δea(s−1)1 = dξa(s−1) − hmξa(s−1),b , δω

a(s−1),b1 = dξa(s−1),b . (5.28)

Once we have (5.27) we can check its integrability to read off the restrictions on dωa(s−1),b1

0 ≡ ddea(s−1)1 = d

(hm ∧ ωa(s−1),m

1

)= −hm ∧ dωa(s−1),m

1 . (5.29)

In principle one should write down all the tensors that can appear on the r.h.s. ofdω

a(s−1),m1 to see which of them passes trough the integrability condition. We are not

going to present this analysis and just claim that the solution has a nice form of

dωa(s−1),b1 = hm ∧ ωa(s−1),bm

1 , a(s− 1), bb = s− 1 , (5.30)

where the new field is a one-form ωa(s−1),bb1 with values in the irreducible representation

(Young and traceless) of the Lorentz algebra specified above. The integrability holds

thanks28 to hm ∧ hn ∧ ωa(s−1),mn1 ≡ 0. A new one-form field comes with the associated

gauge parameter, so the last equation ensures the invariance under

δωa(s−1),b1 = dξa(s−1),b − hmξa(s−1),bm , δω

a(s−1),bb1 = dξa(s−1),bb . (5.32)

Let us emphasize that we found a new field that was absent in the case of gravity! Tohave a vielbein and a spin-connection was enough for a spin-two field. ω

a(s−1),bb1 is the first

27Already this equations as well as all others below, points towards some algebra, whose connectionA contains the gravity sector in terms of vielbein/spin-connection and higher-spin connections, of whichea(s−1) is a particular component. What we see is the linearization of F = dA+ 1

2 [A,A] under A0 + gA1,which gives F = dA+ [A0, A], A0 = h,.

28The following trivial identities are used here and below

ha ∧ hb + hb ∧ ha ≡ 0 , hm ∧ hm ≡ 0 . (5.31)

43

of the extra fields that appear in the frame-like formulation of higher-spin fields, [3, 95].The necessity for the extra fields can be seen already in (5.27). From the pure gravitypoint of view this is a torsion constraint, which allows one to express spin-connection asa derivative of the vielbein. This is not so for s > 2. One can either find that spin-connection ω

a(s−1),b1 cannot be fully solved from (5.27) or observe that (5.27) is invariant

under δωa(s−1),b1 = hmξ

a(s−1),bm. These are equivalent statements as the component ofthe spin-connection affected by the extra gauge symmetry cannot be solved for. Theappearance of an additional symmetry tells us there is an associated gauge field, the firstextra field. The extra gauge symmetry can also be seen in the quadratic action built fromea(s−1)1 and ω

a(s−1),b1 , [94].

At this stage it is necessary to see if the Fronsdal equations appear inside (5.30). It is

a right time for them to emerge since ωa(s−1),b1 is expressed as ∂ea(s−1) ∼ ∂φa(s) via (5.27)

and (5.30) contains ∂∂φa(s). The Fronsdal equations are imposed by the same trick as ingravity — the r.h.s. of (5.30) parameterizes those derivatives that can be non-zero on-shell. We postpone this check till the summary section. Once we have found the secondequation we can check its integrability. This process continues smoothly giving

dωa(s−1),b(k)1 = hm ∧ ωa(s−1),b(k)m

1 , a(s− 1), b(k) =ks− 1 , (5.33)

until

dωa(s−1),b(s−2)1 = hm ∧ ωa(s−1),b(s−2)m

1 , a(s− 1), b(s− 1) = s− 1 , (5.34)

the integrability of which implies hk ∧ dωa(s−1),b(s−2)k1 ≡ 0, the unique solution being

dωa(s−1),b(s−1)1 = hm ∧ hnCa(s−1)m,b(s−1)n , a(s), b(s) = s , (5.35)

where Ca(s),b(s) is a zero-form. It is easy to see that it is a solution. Indeed, hk ∧hm∧hn∧Ca(s−1)m,b(s−2)kn ≡ 0 since two anticommuting vielbein one-forms are contracted with twosymmetric indices, (5.31).

On-shell Ca(s),b(s) is expressed as order-s derivative of the Fronsdal field. It is calledthe (generalized) Weyl tensor for a spin-s field and it coincides with the (linearized) Weyltensor if we set s = 2. Apart from the Fronsdal operator the Weyl tensor is also gaugeinvariant. One can prove that there are two basic gauge invariants, the Fronsdal operatorand the Weyl tensor. The rest of invariants are derivatives of these two.

Again we have the integrability condition for the Weyl tensor,

hm ∧ hn ∧ dCa(s−1)m,b(s−1)n ≡ 0 . (5.36)

This we can easily solve. First, we decompose ∂cCa(s),b(s) into irreducible tensors of theLorentz algebra

s ⊗ = s ⊕s− 1

s ⊕ s(5.37)

All of these can in principle appear on the r.h.s. of dCa(s),b(s) = h.... The third component,call it Ca(s),b(s),c does not pass the integrability test, giving hm∧hn∧hkCa(s−1)m,b(s−1)n,k 6= 0.

44

The first component Ca(s+1),b(s) passes the test hm ∧ hn ∧ hkCa(s−1)km,b(s−1)n ≡ 0. Thereason is simple, three vielbeins form a rank-three antisymmetric tensor and there is notenough room in the tensor having the symmetry of a two-row diagram to be contractedwith it29. It may seem that the second component, Ca(s),b(s−1) passes the test too, this isnot true, however. The reason is that it appears as the trace. Had we been not interestedin traces, the decomposition would have contained two components only, given by the gldtensor product rule,

s ⊗gld = s ⊕s

(5.38)

The point is that both the components in the decomposition above have their traces of thetype Ca(s),b(s−1) and these are the same ∂mC

a(s−1),b(s−2)m. From the so(d− 1, 1) decompo-sition, we know that there is only one trace. The traces are irrelevant for the integrabilitycondition, so if the second component above does not go through the integrability test,so do all its traces, i.e. Ca(s),b(s−1).

Eventually, one is left with Ca(s+1),b(s) and the equation now reads

dCa(s),b(s) = hm

(Ca(s)m,b(s) +

1

2Ca(s)b,b(s−1)m

). (5.39)

Again, Ca(s)m,b(s) alone does not have the symmetry of the l.h.s, hence we have to projectit appropriately by hand, which is done with the help of the second term in the brackets.Proceeding this way we arrive at

dCa(s+k),b(s) = hm

(Ca(s+k)m,b(s) +

1

k + 2Ca(s+k)b,b(s−1)m

), a(s + k), b(s) = ks .

Unfolded equations for any s. Summarizing, we get the following diverse equations

Dω

a(s−1),b(k)1 = hc ∧ ωa(s−1),b(k)c

1 ,

δωa(s−1),b(k)1 = Dξa(s−1),b(k) − hc ξa(s−1),b(k)c ,

0 ≤ k < s− 1 , (5.40)

Dω

a(s−1),b(s−1)1 = hc ∧ hdCa(s−1)c,b(s−1)d ,

δωa(s−1),b(s−1)1 = Dξa(s−1),b(s−1) ,

k = s− 1 , (5.41)

DCa(s+i),b(s) = hc

(Ca(s+i)c,b(s) + 1

i+2Ca(s+i)b,b(s−1)c

),

δCa(s+i),b(s) = 0 ,i ∈ [0,∞) , (5.42)

where we have replaced d with the background Lorentz derivative D = d + , whichmakes the system valid in any coordinate system in Minkowski space. Note that D2 = 0.Let us enlist once again the spectrum of fields (differential form degree and the Young

29It is the property of tensors having the symmetry of one- and two-row Young diagram that anti-symmetrization over any three indices vanishes.

45

shape of the fiber indices)

grade : 0 1 ... s− 1 s s+ 1 ...

Youngshape

: s− 1 s− 1 ... s− 1 s s ...

degree : 1 1 ... 1 0 0 ...,

We recover below the Fronsdal equations from the unfolded ones. What is left aside isthat the most of the fields are not dynamical. Except for those components of the framefield which are in one-to-one correspondence with the Fronsdal field all other componentsare either auxiliary or Stueckelberg. Auxiliary fields are those that can be expressed asderivatives of the Fronsdal field by virtue of the equations of motion, while Stueckel-berg ones can be gauged away with the help of the algebraic gauge symmetry (similarlyto how the extra components of the frame field can be gauged away by local Lorentztransformations). Rigorous proof of these facts requires an advanced technology, calledσ−-cohomology, [95].

Fronsdal equations from unfolded ones. We already know the way the Fronsdalfield is embedded into the frame-like field. Let us now show where the Fronsdal equationsreside. We need the first two unfolded equations with the form indices revealed

∂mea(s−1)n − ∂nea(s−1)

m = hcmωa(s−1),cn − hcnωa(s−1),c

m , (5.43)

∂mωa(s−1),bn − ∂nωa(s−1),b

m = hcmωa(s−1),bcn − hcnωa(s−1),bc

m . (5.44)

Converting all indices to the fiber

ea(s−1)|b ≡ ea(s−1)m hbm, ωa(s−1),b|c ≡ ωa(s−1),b

m hcm, ωa(s−1),bb|c ≡ ωa(s−1),bbm hcm (5.45)

we get

∂cea(s−1)|d − ∂dea(s−1)|c = ωa(s−1),c|d − ωa(s−1),d|c, (5.46)

∂cωa(s−1),b|d − ∂dωa(s−1),b|c = ωa(s−1),bc|d − ωa(s−1),bd|c . (5.47)

By virtue of the first equation the spin-connection can be expressed in terms of the firstderivatives of the frame field which contains the Fronsdal tensor together with a pure gaugecomponents. Then, the second equation imposes the Fronsdal equation and expresses thesecond spin-connection. In order to project onto the Fronsdal equations we contract (5.47)with ηbd and symmetrize over c and a(s− 1), which gives

F a(s) = ∂cωa(s−1),c|a − ∂aωa(s−1),c|

c = 0. (5.48)

Note that the extra field disappeared because of the specific projection made. Now wesymmetrize a(s− 1) and d in (5.46)

ωa(s−1),c|a = ∂cea(s−1)|a − ∂aea(s−1)|c , (5.49)

46

so that we can express the first term in (5.48). To express the second of (5.48) we contract(5.46) with ηda, which gives

ωa(s−1),c|

c =(∂ce

a(s−2)c|a − ∂aea(s−2)c|c

). (5.50)

Plugging the last two equations into (5.48) we find

ea(s−1)|a − ∂a∂c(ea(s−2)c|a + ea(s−1)|c

)+ 2∂a∂ae

a(s−2)c|c = 0 . (5.51)

Now we have to remember how the Fronsdal field is embedded into the frame field

ea(s−1)|a = φa(s) , ea(s−2)c|

c =1

2φa(s−2)c

c , ea(s−2)c|a + ea(s−1)|c = φa(s−1)c . (5.52)

Magically, all the terms come exactly in the combinations above, which leads to theFronsdal equations, (2.4),

φa(s) − ∂a∂cφa(s−1)c + ∂a∂aφa(s−2)c

c = 0. (5.53)

We have seen already that the gauge transformations for the totally symmetric componentof the frame field ea(s−1)|a are those of the Fronsdal field while the extra component canbe gauged away.

Lower-spins. It is obvious that s = 2 is a particular case of the above construction.One simply sets s = 2 to get the unfolded equation describing free graviton of Section4 and 5.1. While one always needs infinitely-many zero-forms, the number of one-formsequals the spin of the field, s. We will see below that lower-spin cases, s = 1 and s = 0are just a degenerate and a very degenerate cases of the spin-s unfolded equations. Inparticular, there are no one-forms in the unfolded formulation of s = 0 in accordance withthe fact that scalar field is not a gauge field. It is worth elaborating on the s = 0, 1 casesfrom scratch.

5.3 s = 0

A free massless scalar field in Minkowski space seems to be the case where the unfoldedapproach gives little advantage. At some point the formulation looks tautological. Theonly reason to consider the scalar case is because it is in this form the scalar field turnsout to be embedded into the full higher-spin theory.

Given a scalar field C(x), it is easy to put the Klein-Gordon equation C(x) = 0 inthe first order form

∂mC(x) = Cm(x) , ηnm∂nCm(x) = 0 = ηnm ∂n∂mC(x) , (5.54)

which is not yet what we need. We can replace an auxiliary field Cm(x) with its fiberrepresentative Ca(x), Ca = hmaCm and rewrite the first equation as

dC = haCa ⇐⇒ dxm (∂mC = ηmaC

a) . (5.55)

47

The difficulty is with the second equation, it must be of the form

dCa = ... (5.56)

where ... means some one-form. We can enumerate all the fields that can appear on ther.h.s. Indeed, l.h.s. reads dxm ∂mC

a. Raising all indices with the inverse vielbein we get∂aCb, i.e. just a rank-two tensor. As such it can be decomposed into three irreduciblecomponents: traceless symmetric – , trace – • and antisymmetric –

= ∂aCa − 1

dηaa∂mC

m , • = ∂mCm , = ∂aCb − ∂bCb . (5.57)

Therefore, introducing Caa, C ′ and Ca,b in accordance with the pattern above, the mostgeneral r.h.s. reads

dCa = hmCam + hmC

a,m +1

dhaC ′ . (5.58)

However, Ca is not an arbitrary fiber vector, it came as a derivative of the scalar andhence ∂aCb − ∂bCa ≡ 0, i.e. Ca,b ≡ 0. By virtue of (5.58) C = C ′. Once C ′ is presenton the r.h.s. there is no Klein-Gordon equation imposed. Therefore, we have to set C ′ tozero too30. Finally, Caa parameterizes those second order derivatives that are generallynon-zero on-shell and the second equation reads

dCa = hmCam . (5.59)

Actually, the absence of Ca,b can be seen in checking the integrability of (5.55). Nilpotencyof the exterior derivative d2 = 0 implies

0 ≡ ddC = d(haCa) = −ha ∧ (hmC

am + hmCa,m +

1

dhaC ′) = −ha ∧ hmCa,m , (5.60)

where we used (5.31). Since the vielbein is invertible we get Ca,m ≡ 0.Can we stop at Caa? By applying d to (5.59) we find that Caa is not unconstrained

0 ≡ ddCa = d(hmCam) = −hm ∧ dCam . (5.61)

So we are led to consider possible r.h.s. in dxm∂mCaa. Decomposing the tensor product

⊗ with and representing Caa and ∂m, respectively, one finds

⊗ = ⊕ ⊕ , (5.62)

i.e. the most general equation for Caa reads

dCaa = hmCaam + hmC

aa,m +

(haCa − 2

dηaahmC

m

). (5.63)

30Another option is to express C′ as a function of C. For example, C′ = m2C, gives Klein-Gordonequation with a mass term. One can also introduce an arbitrary potential C′ = V (C).

48

Note that haCa is not traceless and has to be supplemented with the second term toagree with vanishing trace of Caa. Both the hook component Caa,b and the trace Ca

are inconsistent with (5.61) since

hn ∧ dCan = hn ∧ hmCan,m − d− 2

2ha ∧ hmCm 6= 0 . (5.64)

Therefore we have to exclude Caa,b and Ca from the r.h.s.. The only term left parame-terizes third-order derivatives of the scalar field on-shell

dCaa = hmCaam . (5.65)

Continuing this way one one arrives at the final answer. A typical situation in obtainingthe unfolded equations is that one makes use of dynamical equations at first few levelswhich define the pattern of first auxiliary fields and then at each next level there are onlyBianchi identities that further constraint the final form of equations.

Unfolded equations for s = 0. The full system of equations that describe a freemassless scalar field reads

DCa(k) = hmCa(k)m , Ca(k−2)mnηmn ≡ 0 . (5.66)

where the set of fields consists of totally-symmetric traceless zero-forms Ca(k).A field Ca(k) with k > 0 is expressed as order-k derivative ∂a...∂aC of the lowest field

C(x) which can be identified with the original scalar field we started with. All derivatives∂a...∂aC are traceless since ηmn∂

mna(k−2)C = ∂a(k−2)C = 0, which explains why Ca(k)

has to be traceless. The Klein-Gordon equation appears thanks to the vanishing trace ofCaa. Simply, the second equation, with Ca already solved as ∂aC, implies ∂a∂aC = Caa.Since the r.h.s. has vanishing trace, by contracting with ηaa we derive C = 0.

In the table below we list the spectrum of fields needed. The form degree is alwayszero.

grade : 0 1 2 3 ...

Young-shape : • ...

degree : 0 0 0 0 ...

This is what one gets upon setting s = 0 in (5.40)-(5.42) and dropping tensors with(s− 1) = −1 indices.

Scalar is almost tautological. In this simple case of a scalar field there is a wayto make the unfolded system tautological. Let us contract all the fiber indices with anauxiliary variable ya to build C(y|x)

C(y|x) =∑

k

1

k!Ca(k)(x) ya...ya (5.67)

49

then (5.66) is rewritten as

(D − hm ∂

∂ym

)C(y|x) = 0 ,

∂

∂ym∂

∂ymC(y|x) ≡ 0 , (5.68)

where the last condition makes Ca(k) traceless, see Appendix E.3. Going to Cartesiancoordinates where there is no difference between the world and fiber indices we find

(∂

∂xm− ∂

∂ym

)C(y|x) = 0 ,

yC(y|x) ≡ 0 . (5.69)

The first equation tells us that the dependence on x is exactly the dependence on y,the solution being C(y|x) = C(x− y), while the second condition imposes Klein-Gordonequation in the fiber. Therefore, the fiber Klein-Gordon equation is mapped to the x-space Klein-Gordon equation. If we switch to a massive scalar or put a scalar on AdSinstead of Minkowski or proceed to fields with spin the correspondence gets far from beingtautological!

It is certainly true that one can write a tautological system that imposes the originalequations in the fiber and then translates the dynamics to the base in a trivial way, likeabove ∂x = ∂y. The point is that such system will not have an unfolded form apart froma simple example above. The reason as we will see is that the unfolded system containsmore fields, especially connections, which brings in the geometry.

5.4 s = 1

The starting point is the gauge transformation law

δAm = ∂mξ , (5.70)

which suggests to treat Am as a one-form A1 = Amdxm, which is what usually done. The

gauge parameter is then a zero-form and we can simply write δA1 = dξ. The equationshave to start with dA = ... and there are three options for what ... might be.

dA1 = R2 , dA = hm ∧ ωm1 , dA = hm ∧ hnCm,n , (5.71)

where R2 is a two-form; ωm1 is some vector-valued one-form and Cm,n is a zero-form that

is antisymmetric in its fiber indices, thus belonging to . Note that these three optionspartially overlap as all of the fields on the r.h.s. contain , whenever all the world indicesare converted to the fiber with the inverse vielbein. No matter how natural the first optionis, it fails. Indeed, the first equation is invariant under δA1 = dξ + ξ1, δR2 = dξ1 whereξ1 is some scalar one-form, the gauge parameter for R2. The latter means that one cangauge away A1 completely with the help of ξ1, which is not what we need. The secondoption fails in a similar way once we observe that there is an additional gauge symmetryδA1 = −hmξm, δωa

1 = dξa with a vector-valued zero-form ξa. Therefore, we have to usethe third option

dA1 = hm ∧ hnCm,n , (5.72)

50

which is just a way to parameterize the Maxwell tensor, ∂mAn−∂nAm = Ca,bhamhbn. Theintegrability of this equation implies

0 ≡ ddA = d(hm ∧ hnCm,n) = hm ∧ hn ∧ dCm,n (5.73)

Again, as in the scalar case, dCm,n is restricted. The most general r.h.s. of dCm,n is givenby evaluating the tensor product ⊗ , where and correspond to ∂a and Ca,b

⊗ = ⊕ ⊕ , (5.74)

= ∂cCa,b − ∂[cCa,b] +2

d− 1ηc[a∂dC

b,]d , (5.75)

= ∂[cCa,b] , (5.76)

= ∂mCa,m = Aa − ∂a∂mAm , (5.77)

so introducing Caa,b, Ca,b,c and Ca in accordance with the pattern above dCa,b reads

dCa,b = hm

(Cam,b +

1

2Cab,m

)+ hmC

a,b,m +(haCb − hbCa

). (5.78)

The terms in brackets make up the rank-two antisymmetric fiber tensor31. We see thatthe presence of the antisymmetric component on the r.h.s. violates integrability as hm ∧hn ∧ hkCm,n,k 6= 0. The same time we know that ∂[mCn,r] = ∂[m∂nAr] ≡ 0, which is theBianchi identity, and hence there is no such component in the jet of Am.

We would like to impose the Maxwell equations, but we see that keeping Ca on ther.h.s. we rather get Aa − ∂a∂mA

m = Ca. Therefore, Ca has to be set to zero. Theleftover field Caa,b with the symmetry of parameterize the second order derivatives ofAm that do not vanish on-shell. Now the second equation in the hierarchy reads

dCa,b = hm

(Cam,b +

1

2Cab,m

). (5.79)

The integrability condition for this equation implies that Caa,b cannot be arbitrary. Astraightforward analysis shows that we need to introduce fields Ca(k),b that have thesymmetry of

kand are traceless.

Unfolded equations for s = 1. The full system reads

DA1 = hm ∧ hnCm,n , δA1 = Dξ , (5.80)

DCa(k),b = hm

(Ca(k)m,b +

1

k + 1Ca(k)b,m

), (5.81)

31Let us note that Cam,b is not antisymmetric in a, b! The relative coefficient is fixed in checking thatthe symmetrization over a, b does vanish and using the Young property to derive Cam,a = −Caa,m.

51

where let us stress again that all fields are so(d − 1, 1)-irreducible as fiber tensors. Itis useful to list the fields in order of their appearance by assigning certain grading. Ascompared to the scalar case we have more diverse structure of fiber tensors and there isone field which is not a zero-form

grade : 0 1 2 3 ...

Young shape : • ...

degree : 1 0 0 0 ...

Again, this is the specialization of (5.40)-(5.42) to s = 1.

5.5 Zero-forms

The first observation is that the equations for zero-forms Ca(s+k),b(s), (5.42), are self-consistent and do not require the presence of any one-forms. What do they describe? Theanswer can be read off from the spin-one case. The zero-forms begin with the Maxwelltensor Ca,b. Since we do not supplement it with gauge potential A1 and the correspondingequation (5.72), it is the fundamental field now. Then (5.79) implies the following twoequations

∂aCb,c + ∂aCb,c + ∂aCb,c = 0 , ∂mCa,m = 0 . (5.82)

Indeed, the most general r.h.s. of (5.79) could involve three components according to(5.74), but it contains only one, (5.79). Therefore, the two combinations of first derivativesof Ca,b are set to zero by (5.79), giving rise to the equations above. These are the standardequations that impose dF = 0 and ∂mF

n,m = 0 for two-form Fm,ndxm∧dxn, Fm,n = Cm,n.

The first equation is the Bianchi identity which implies F = dA for some A, so thegauge potential is implicitly there. The second one is the Maxwell equations without anysources.

The same story for the spin-two case gives the Weyl tensor Caa,bb as the fundamentalvariable and imposes

∂[aCbc],de = 0 , ∂mCab,cm = 0 , (5.83)

which can be recognized as the two components of the differential Bianchi identities forthe Weyl tensor.

In the case of a spin-s field we find the following equations

∂[uCa(s−1)u,b(s−1)u] = 0 , ∂mCa(s),b(s−1)m = 0 , (5.84)

where the anti-symmetrization over three u’s is implied. The rest of equations merelyexpress the fields as derivatives of the generalized Weyl tensor. The simplest way to solveequations is to perform Fourier transform to find

Ca(s),b(s)(p) = ϕa(s)(p) pb...pb + permutations , (5.85)

52

where ϕa(s)(p) is a symmetric tensor that depends on momentum pa and the followingadditional conditions are true

pmpm = 0 , ϕa(s−2)m

m = 0 , pmϕa(s−1)m = 0 . (5.86)

We almost reproduced the on-shell conditions (2.10)-(2.12) for the Fronsdal field. Whereis the gauge symmetry? The Young properties imply that ϕa(s) of special form paξa(s−1)

yields a vanishing Weyl tensor. This is the on-shell Fronsdal gauge transformations in mo-mentum space. It is not possible to build an antisymmetric tensor with papb, which is au-tomatically symmetric. Likewise, it is not possible to build Ca(s),b(s) out of pb...pbpaξa(s−1).The problem is that two p’s show up in the same column of the Young diagram, papb.By going into antisymmetric presentation, where Ca(s),b(s) has s pairs of antisymmetricindices, Cab,cd,..., we see that the gauge transformation of φa(s) does not affect the Weyltensor. Therefore, φa(s) is defined up to a gauge transformation and we recover the on-shellFronsdal theory.

The upshot is that the equations for zero-forms also describe a spin-s massless field,but in a non-gauge way. The examples of s = 1, 2 tells us that it is the gauge potentialsA and gmn (which needs to be replaced by ea, ωa,b) that mediate interactions. Therefore,while the non-gauge description of massless fields is perfectly fine at the free level it iscrucial to supplement it with gauge potentials in order to proceed to interactions. Thefull unfolded equations tells us that the gauge potentials are determined up to a gaugetransformation by the gauge invariant Weyl tensors (field-strengths).

5.6 All spins together

One can write a relatively simple system that describes fields of all spins s = 0, 1, 2, 3, ...together. This is a Minkowski prototype of the unfolded equations in AdS4 of Sections 9and 10. First we sum up all the fields that appeared above into generating functions oftwo auxiliary variables, ya, pa,

ω1(y, p|x) =∑

s>0

k=s−1∑

k=0

1

(s− 1)!k!ωa(s−1),b(k)1 ya...ya pb...pb , (5.87)

C(y, p|x) =∑

s=0

∑

k=0

1

(s+ k)!s!Ca(s+k),b(s)ya...ya pb...pb . (5.88)

Analogous decomposition is used for the collective gauge parameter ξ(y, p|x). One of the

crucial observations is that one-forms ωa(s−1),b(k)1 when summed over all spins, cover all

irreducible Lorentz representations with the symmetry of all two-row Young diagrams,each appearing in one copy

ωa(s−1),b(k)1 a(s− 1), b(k) :

ks− 1 (5.89)

The zero-forms cover the same variety of Young diagrams (all two-row)

Ca(s+k),b(s) a(s+ k), b(s) :ss+ k (5.90)

53

Therefore, the fiber spaces of one-forms and zero-forms are isomorphic!We need the Taylor coefficients of (5.87)-(5.88) to obey Young symmetry and trace

constraint, which implies32

yn

∂

∂pn,

∂2

∂ym∂ym,

∂2

∂ym∂pm,

∂2

∂pm∂pm

(ω1(y, p|x), C(y, p|x), ξ(y, p|x))≡ 0 (5.91)

Then the unfolded system of equations together with the gauge transformations can berewritten as

Dω1(y, p|x) = δNy ,Nphm ∧ hn ∂2

∂ym∂pnC(y, p|x) , (5.92)

δω1(y, p|x) = Dξ(y, p|x) , (5.93)

DC(y, p|x) = 0 , (5.94)

D = D − hm ∂

∂pm, (5.95)

D = D − hm(

∂

∂ym+

1

Ny −Np + 2pn

∂

∂yn∂

∂pm

), (5.96)

D = d+a,b

(ya

∂

∂yb+ pa

∂

∂pb

), (5.97)

where Ny = ym∂/∂ym, Np = pm∂/∂pm are Euler operators that just count the numberof tensor indices, i.e. [Ny, ya] = +ya, [Ny, ∂/∂y

a] = −∂/∂ya, etc. The Lorentz-covariantderivative, D, must hit every index contracted with ya or pb. This is achieved with thehelp of the differential operator in the last line. In fact, it makes use of the standardrepresentation of the orthogonal algebra on functions

a,b

(ya

∂

∂yb

)=

1

2a,b

(ya

∂

∂yb− yb

∂

∂ya

). (5.98)

Taylor expanding this system we find all the unfolded equations that we derived above.The Kronecker symbol δNy ,Np ensures that only the Weyl tensors of C contribute to ther.h.s. of the equations for ω. One of the effective realization of δNy ,Np could be

C(yt, pt−1)∣∣t=0

(5.99)

The system is gauge invariant and consistent thanks toD2 = 0, D2 = 0. It is importantthat the term that sources one-forms ω by zero-forms C does not spoil the consistency.The system can be projected onto particular spin-s subsector with the help of

Nyω1(y, p|x) = (s− 1)ω1(y, p|x) , NpC(y, p|x) = sC(y, p|x) . (5.100)

The system describes free fields of all spins s = 0, 1, 2, 3, ... over the Minkowski space. Itis a very nontrivial problem to find a nonlinear theory that leads to this equations in the

32See Appendix E.3 for the implementation of Young-symmetry/trace constraints in terms of generatingfunctions. It is these constraints that will be shown to be automatically solved with the help of spinorialauxiliary variables later on, which makes d = 3, 4 HS theories simpler.

54

linearization. For this case it was proved, [96], under certain assumptions, that it is notpossible to find a nonlinear completion for the system above. But it becomes possible inanti-de Sitter space.

There is even a simpler set of equations, [96], that does the same with the help ofadditional auxiliary fields.

6 Unfolding

In this section we would like to give a systematic overview of the unfolded approach, [40,41], which underlies the Vasiliev HS theory. The unfolded approach is a universal methodof formulating differential equations on manifolds. Any set of differential equations can beput into this form in principle. Once it is done one can learn a lot about the symmetriesof the original system. The symmetries that were not at all obvious in the originalformulation become manifest when the unfolded form of equations is available. Thisturned out to be very useful for the HS problem analysis since the HS theory is so muchconstrained by the symmetry.

The unfolded approach is one of the general methods and as such it is not a cure-all.It can provide some substantial progress once there is a rich hidden symmetry behindthe system, but it may not give any advantage as compared to other approaches whendealing with systems that have a poor symmetry. Besides, the unfolding of some nonlineardynamical equations can be of a great technical challenge let alone there are only fewexamples of nonlinear equations in the literature that acquire explicit unfolded form.

Few of the nice features of the unfolded approach include: its manifest diffeomorphisminvariance; it is specifically suited to account for gauge symmetries, conserved currentsand charges. Generally when looking for the interactions starting from a linear gaugetheory, one is faced with several problems at once: find a deformation of Lagrangian orequations of motion and a deformation of the gauge symmetry in such a way as to leaveLagrangian or equations of motion invariant. These deformations are governed by thesame set of structure constants within the unfolded approach.

The variety of unfolded equations is at least as rich as all Lie algebras together withall representations thereof and cohomologies. As we will see all of the structure constantsthat are at the bottom of any unfolded equations have certain interpretation within theLie algebra theory.

Though at present it is far from being clear, the unfolded approach have a potentialto be the generalization of the notion of integrability in higher dimensions. Certainaspects of the unfolded approach showed up in the context of supergravity [97–99] and intopology [100] under the name of Free Differential Algebras.

6.1 Basics

LetWA be a set of differential forms over a certain manifoldMd. The index A is a formalone that we let run over some set of fields assuming the Einstein summation convention.

55

In practice it runs over a set of linear spaces33. The differential form degree of WA isdenoted by |A|. Equations of the following form are referred to as the unfolded equations

dWA = FA(W ), (6.1)

where d is the exterior (de-Rham) differential and the function FA(W ) has degree |A|+1and is assumed to be expandable in terms of the exterior products of the fields themselves

FA(W ) =∑

n

∑

|B1|+...+|Bn|=|A|+1

fAB1...Bn

WB1 ∧ ... ∧WBn, (6.2)

The structure constants fAB1...Bn

are certain elements of Hom(B1⊗ ...⊗Bn,A), i.e. mapsfrom a tensor product B1 ⊗ ...⊗Bn to A. They satisfy the symmetry condition inducedby the form degree of the fields

fAB1...BiBi+1...Bn

= (−)|Bi||Bi+1|fAB1...Bi+1Bi...Bn

. (6.3)

The crucial point is to impose the integrability condition

FB ∧ δFA

δWB≡ 0, (6.4)

that arises when applying d to both sides of (6.7), using d2 = 0 and the equations ofmotion again on the r.h.s. (we will use ∂B instead of δ

δWB )34

0 ≡ d2WA = dFA(W ) = dWB ∧ ∂BFA(W ) = FB ∧ ∂BFA(W ) . (6.5)

The integrability condition implies that the equations are formally consistent and containall their differential consequences35. In deriving (6.4) we also assume the absence ofmanifest space-time dependence in FB.

When written in terms of the structure constants the integrability condition reads∑

fCD1...Dm

fACB2...Bn

WD1 ∧ .... ∧WBn = 0 , (6.6)

and reminds the Jacobi identity. Let us note that some of these equations may hold auto-matically if the total differential form degree exceeds the dimension of the manifoldMd.The useful concept introduced by Vasiliev is that of the universal unfolded system, [65,96].The unfolded equations are called universal iff the system is integrable irrespectively ofthe dimension, i.e. formally for any d. In other words, the integrability constraint in somefixed dimension d = d0 may admit nontrivial solution due to the fact that any differentialform of the degree d0+1 is identically zero. The universal unfolded systems are supposedto be consistent for any d.

The bonus of the universal system, which has been already shown to be useful inapplications, is that it remains meaningful on a manifold of any dimension. In particular

33For example, A runs over Lorentz tensors with the symmetry of ks−1 , k = 0, ..., s−1 and s

s+n ,n = 0, 1, ... in the case of a spin-s field as we have seen in Section 5.

34The derivatives carry the same grading as the fields, i.e. ∂A∂B = (−)|A| |B|∂B∂A.35Recall footnote 12 on page 20.

56

one can consider the same unfolded equations over different space-times (say, AdSd+1 andits conformal boundary Md, which could be a way to make AdS/CFT tautological [101]).

As the unfolded equations are formulated entirely in the language of differential forms,there is no need for extra data, like metric or Christoffel symbols, that Md is typicallyequipped with.

It is convenient to rewrite the unfolded equations as the zero-curvature condition

RA = 0 , RA ≡ dWA − FA(W ) , (6.7)

where RA will be referred to as the curvature.The unfolded equations turn out to implement the concept of gauge symmetry auto-

matically. Indeed, let us define the following gauge variations

δǫWA = dǫA + ǫB∂BF

A, if |A| > 0, (6.8)

δǫWA = +ǫB

′

∂B′FA, B′ : |B′| = 1, if |A| = 0, (6.9)

where we have to distinguish between forms of degree greater than zero and zero-forms.Each WA with |A| > 0 has an associated gauge symmetry with parameter ǫA that is adegree |A| − 1 form taking values in the same space as WA does. There are no genuinegauge symmetries associated with zero-forms. They transform without the dǫA-piecesimilar to matter fields and with the parameters of the degree-one fields among WA theyare coupled to.

One can check that the variation of the curvature RA vanishes on-shell, i.e. whenRA = 0, since

δRA = (−)C+1ǫCRB∂B∂CFA . (6.10)

This is a generalization of δF = [F, ǫ] in the Yang-Mills theory with ∂B∂CFA playing the

role of the structure constants of the Lie algebra, which in general are not constants anddo depend on the fields.

The integrability condition can be rewritten as a Bianchi identity for the curvature

dRA +RC∂CFA ≡ 0 , (6.11)

which is a generalization of DF ≡ 0 in the Yang-Mills.There is also a generalization of the fact that within a gauge theory a diffeomorphism

can be represented as a gauge transformation plus a correction due to nonvanishing cur-vature

LξA = D(iξA) + iξF (A) , F (A) = dA+1

2[A,A] . (6.12)

The analog within the unfolded approach reads

LξWA = δiξ·WW

A + iξ · RA (6.13)

the difference is that within the unfolded approach any diffeomorphism is always a par-ticular gauge transformation on-shell, since RA = 0.

The notion of reducible gauge symmetries is also easily implemented. Indeed, thecondition for WA to be invariant under the gauge transformation

δǫWA = 0 , dǫA − ǫB∂BFA = 0 , (6.14)

57

can be interpreted as the unfolded-like system itself with the field content extended bythe gauge parameters ǫA. As such it is gauge invariant under the second-level gaugetransformations

δǫA = dξA + ξB∂BFA , (6.15)

where we neglect terms bilinear in ǫA by the definition of what gauge transformationsare. Again, δǫA = 0 is an unfolded equation and so on until the zero-forms are reached.The level-two gauge parameter here ξA is a degree-(|A| − 2) form. Therefore, a degree-qfield has q levels of gauge (and gauge for gauge) transformations in total.

6.2 Structure constants

In this section we unveil the algebraic meaning of the structure constants. As we will seesoon, the sector of one-forms is at the core of any unfolded-system as they necessarilybelong to some Lie algebra, say g. The rest of the fields can be interpreted as takingvalues in various g-modules. These modules can be in general glued (coupled) togethervia Chevalley-Eilenberg cocycles of g.

Lie algebras and flatness/zero-curvature. Suppose there are one-forms only or aclosed subsector thereof. Let ΩI ≡ ΩI

µdxµ denote the components of the one-forms in

some basis. Then the only unfolded-type equations one can write read

dΩI +1

2f IJKΩ

J ∧ ΩK = 0, (6.16)

where f IJK are the structure constants that are antisymmetric in J,K since one-forms

anti-commute. The integrability condition (6.4) implies the Jacobi identity

f IJKf

JLMΩK ∧ ΩL ∧ ΩM ≡ 0 ←→ f I

J [KfJLM ] ≡ 0 . (6.17)

Therefore, f IJK define certain Lie algebra, call it g. Then (6.16) is the flatness or zero-

curvature condition for a connection of g. Equivalently, (6.16) means that the Yang-Millsfield-strength F = dΩ+ 1

2[Ω,Ω] vanishes. The gauge transformations associated with the

zero-curvature equations are the standard Yang-Mills transformations, δΩI = DΩǫI . Such

equations describe background geometry, e.g. Minkowski or AdS, (3.54).

Modules and covariant constancy. Consider now the extension of the unfolded sys-tem with one-forms by a sector WA

q of q-forms. The most general equations that arelinear in WA

q read

dWAq + fI

AB Ω

I ∧W Bq = 0 , (6.18)

with ΩI obeying (6.16) and fIAB being some new structure constants. The integrability

(6.4) then implies

ΩJ ∧ ΩK

(−12f IJKfI

AB + fJ

ACfK

CB

)W B

q = 0 , (6.19)

58

where there is an implicit anti-symmetrization over J,K in the last term due to anti-commuting nature of ΩI . Thinking of fI

AB as endomorphisms fi ∈ Hom(V, V ), just

matrices on the space V where the q-forms take value, we recognize the definition of ag-module

fJ fK − fK fJ = [fJ , fK ] = f IJKfI (6.20)

where denotes the usual matrix product over implicit indices A.Therefore, writing down (6.18)-type equations is equivalent to specifying certain rep-

resentation V of g and the corresponding equation is the covariant constancy conditionfor a field with values in V ,

DΩWAq ≡ dWA

q + fIABΩ

I ∧W Bq = 0 , δWA

q = DΩξAq−1 , (6.21)

where we added gauge transformations — the covariant derivative in the module definedby fI

AB. The gauge invariance is thanks to DΩDΩ = 0, (6.16).

Contractible cycles and empty equations. Consider now the linear equations ofthe form

dWAq = fA

BWB

q+1. (6.22)

By linear transformations they can always be split into two types of subsystems fAB

I : dWA

q +WA

q+1 = 0, dWB

q+1 = 0, (6.23)

II : dW A

q = 0, (6.24)

where the last equation of (6.23) is the integrability condition for the first one. (6.23)is called a contractible cycle, [100]. With the help of gauge transformations δWA

q =dξAq−1 − χA

q we can always gauge away WAq since χA

q has the same number of componentsand enters algebraically. Then we are left with WA

q+1 = 0. Consequently we see that theequations of type (6.23) are dynamically empty. The type-II equations can be solved ina pure gauge form W A

q = dξA by virtue of the Poincare lemma unless q = 0, see extraSection 12.7. If q = 0 the solution is just a constant, cA, which cannot be gauged away.Therefore, the equations of type (6.24) are dynamically empty unless q = 0. In the lattercase the dimension of the solution space is that of W A

0 at a point, i.e. cA.In what follows we will never meet contractible cycles as parts of the unfolded equa-

tions. Clearly their presence is redundant or at least unnecessary. Being found in someunfolded system, such contractible subsystems can always be removed. By contrast, type-II for q = 0 and with d extended to the covariant derivative in some module will be shownto be very important since they carry all degrees of freedom.

Cocyles and couplings. Here we consider a more and the most general types of un-folded equations that can appear, while somewhat technical discussion is left to extraSection 12.4. At the free level all equations are linear in matter fields described by someWA

p , WA

q , etc, but they can be nonlinear in the background fields, i.e connection Ω,

DΩWAp ≡ dWA

p + fIAB Ω

I ∧W Bp = fI1...Ik

ABΩI1 ∧ ...ΩIk ∧WB

q , (6.25)

DΩWA

q = ... (6.26)

59

As it is explained in extra Section 12.4, fI1...IkABis a Chevalley-Eilenberg cocycle of the

underlying Lie algebra, g, where Ω takes values in. The cocycle takes values R1 ⊗ R∗2,

where g-modules R1 and R2 are associated with WAp and WA

q , respectively.Unfolded equations that have something to do with interactions have to be nonlinear.

Assuming that there is a well-defined linearization, i.e. nonlinearities can be neglected insome limit, we should get the structures of the form (6.25) at the free level. This equipsus with a bunch if g-modules, R1, R2, ... and Chevalley-Eilenberg cocyles. Nonlineardeformations of (6.25) can have the following form

DΩWAp ≡ dWA

p + fIAB Ω

I ∧W Bp = fI1...Ik

AB1...Bm

ΩI1 ∧ ...ΩIk ∧WB1q1∧ ... ∧WBm

qm. (6.27)

Again, fI1...IkAB1...Bm

is a Chevalley-Eilenberg cocycle. The gauge transformations canalso be written down.

Zero-forms and degrees of freedom. The sector of zero-forms is a distinguished one.First of all, the most general linear equations in the sector of zero-forms, say CA, havethe form of the covariant constancy condition

dCA + fA

IB ΩI ∧ CB = 0 (6.28)

and no sources on the r.h.s. are allowed by the form-degree argument. Indeed, the sourcesmust be at most and at least linear in Ω, which means that the sources are zero-forms, sowe can move them on the l.h.s. Therefore, zero-forms are initial objects in a sense thatthey can source forms of higher degrees, but nothing can source zero-forms.

The most important property of zero-forms is that they are moduli of local solutions.Locally, by Poincare Lemma one can always solve dω = 0 as ω = dξ unless ω is a zero-form. For zero-forms dCA = 0 has the solution space, CA(x) = cA = const, that isisomorphic to the space where CA(x) takes values in, say R. The analogous statementon the size of the solution space can be made about the covariant analog DΩC

A = 0 ofdCA = 0. The covariant generalization of Poincare Lemma implies that all fields that areforms of non-zero degree can be represented locally as pure gauge unless they are sourcedby zero-forms, in the later case they consist of two pieces, pure gauge and another onedetermined by zero-forms. Zero-forms are not pure gauge. There are no gauge parametersat all associated with zero-forms. Therefore, locally a solution of any unfolded system isdetermined by the values of all zero-forms at a point, i.e. R, up to a gauge transformation.

That is why we find an infinitely many zero-forms when unfolding fields of differentspins. Fields carry infinitely many ’degrees of freedom’ in a sense that the solutions offield equations are parameterized by functions on Cauchy surface (usually it is the numberof functions referred to as the number of physical degrees of freedom) and these functionsare equivalent to infinitely many constants parameterized by the zero-forms at a point,cA.

Combining together the fact that locally solution is reconstructed from values of zero-forms at a point and the general property of unfolded equations that zero-forms take valuesin certain representation, say R, of the underlying Lie algebra, we can conclude that R isrelated to the space of one-particle states, [102, 103]. Remember that the solution spaceof field equations (one particle states) must carry a unitary irreducible representation, say

60

R′ of the space-time symmetry algebra. Since the solution is parameterized by zero-forms,it is actually parameterized by R. So R and R′ must be tightly related. In fact, they areequivalent when complexified.

Let us finish with the following illustration of the unfolded approach. Up to the issuesrelated to gauge symmetries and the fact that unfolded equations are diffeomorphisminvariant and could be nonlinear the idea is to identify those derivatives of original dy-namical fields that remain non-zero on-shell. Imposing equations of motions, Eq(φ) = 0sets some of the derivatives ∂kφ to zero, the rest is then parameterized by zero-forms.Roughly speaking, at every order ∂kφ there remains just one component, but k runs fromzero to infinity.

Figure 2: The idea of the unfolded approach.

φ

∂kφ

Eq(φ) = 0, δφ = ∂ξ

degrees of freedom,i.e. zero-forms

derivatives

Unfolded systems of HS type. In Section 5.6 we found that the linearized unfoldedequations that describe fields of all spins read schematically as

dΩI + f IJKΩ

J ∧ ΩK = 0 , (6.29)

dωA + fAJB Ω

J ∧ ωB = fAJK|B Ω

J ∧ ΩK ∧ CB , (6.30)

dCA + fAIB Ω

I ∧ CB = 0 , (6.31)

where ΩI = ha, a,b is the Poincare connection. Recall, see Section 5.6, that thespace of one-forms ωA, when summed over all spins, is isomorphic to that of zero-forms.They cover irreducible Lorentz representations with the symmetry of all two-row Youngdiagrams, each appearing once. That is why we used the same index for ωA and CA. Thecocycle on the r.h.s. of (6.30) was found to have the form

hm ∧ hnCa(s−1)m,b(s−1)n . (6.32)

61

The problem of interactions can be reduced to seeking for the nonlinear terms suchthat the integrability is preserved. The form degree argument shows that nonlinearitiescan be of two types: fA

BC ωB ∧ ωC on the l.h.s. of (6.30) and zero-forms can source r.h.s.

of all equations by the terms of the form ω ∧ ωCC...C, (6.30), and ωCC...C, (6.31).Already at this point we can see that there must be an infinite-dimensional Lie algebra,

g, of which ωA are gauge fields. Recall that ΩI is a subset of ωA associated with thegravity sector. At the linearized level we cannot fully see this algebra. We found apart of structure constants. Namely, f I

JK , which define the symmetry algebra, h, of thebackground space and fA

JC, which show g as a module under its subalgebra h, but notall fA

BC. Therefore, the first question is the existence of such an algebra and secondlywhether nonlinear deformations of equations are possible or not. To solve the problemwithin the perturbation theory one has to switch on the cosmological constant, whichreplaces h = iso(d − 1, 1) with so(d − 1, 2) or so(d, 1). Then all questions above can beanswered in affirmative and the solution is given by the Vasiliev equations. It is importantthat the spectrum of the fields we found does not change when the cosmological constantis turned on. Like in the Fronsdal theory, there are some corrections proportional to Λ.Schematically the solution reads

dωA + fABC ω

B ∧ ωC = fABC|D ω

B ∧ ωC ∧ CD + fABC|DE ω

B ∧ ωC ∧ CD ∧ CE + ... , (6.33)

dCA + fABC ω

B ∧ CC = fAB|CD ω

B ∧ CC ∧ CD + ... , (6.34)

where we stress that zero-forms and one-forms take values in the same linear space, whosebasis is given by irreducible Lorentz tensors with the symmetries of all two-row Youngdiagrams, each in one copy.

As was noticed in [41], the unfolded equations for higher-spin fields have the followingstructure

dωA = FA(ω,C) , (6.35)

dCA = CB ∂

∂ωBFA(ω,C) , (6.36)

i.e. the equations for zero-forms are built from the same structure function FA(ω,C) asthe equations for one-forms. Formally, we applied CB ∂

∂ωB to the first equation in order toget the second one. For example, this operation transforms dω+ 1

2[ω, ω] into36 dC+[ω,C].

So, there is no doubling of structure constants, (6.34) is fully determined by (6.33). Letus note that (6.36) is nevertheless not an integrability consequence of (6.35).

Summary. We found that the structure constants of any unfolded system of equationshave certain interpretation in terms of Lie theory. There is some underlying Lie algebra,which is related to the subsector of one-forms. The rest of the fields take values in certainmodules of this Lie algebra. Couplings between different sectors of forms are given bycertain representatives of the Chevalley-Eilenberg cohomology groups. Locally, solutionto any unfolded system is reconstructed up to a gauge transformation from knowing thevalues of all zero-forms at some point.

36There is a technical detail related to fact that C takes values in the twisted-adjoint representation,which will become clear around (10.47) and it does not violate the statement that the equations for C

are determined by those for ω.

62

7 Higher-spin theory in four dimensions

In the remaining we will consider the application of unfolded approach to four dimensionalHS theory. Remarkable progress has been achieved in lower dimensions d = 3 and d = 4due to effective twistor-like language available in these dimensions. It turns out to beperfectly suited for dealing with HS algebras and eventually brings the one to nonlinearHS equations. We restrict ourselves to the case of four dimensions, for d = 4 unlike d = 3case admits propagation of HS fields. To proceed in this direction we need to reformulateour settings in terms of spinor language.

8 Vector-spinor dictionary

In dealing with irreducible tensors we have to preserve Young symmetry properties andthe tracelessness. The advantage of the spinorial language as compared to the tensor one isthat the trace and Young symmetry constraints get automatically resolved. Unfortunately,it is available in specific dimensions only. It is the condition for traces to vanish that leadsto complicated trace projectors, the Young symmetry is not for free though too. We havenot faced any trace projectors yet since we developed a theory in Minkowski space. Theunfolded equations for higher-spin fields in AdS get modified by more complicated termswhich are not seen when Λ = 0, e.g.

DCa(k),b(l) = ... + Λ

(haCa(k),b(l) − k

(d+ 2k − 2)hmC

a(k−1)m,b(l)ηaa−

− l

d+ k + l − 3hmC

a(k),mb(l−1)ηab+ (8.1)

+k

l(d+ 2k − 2)(d+ k + l − 3)hmC

a(k−1)b,b(l−1)mηaa),

getting more and more involved at each level of algebraic manipulations. The unfoldedequations for the complete multiplet of higher-spin fields have a simpler form, but stillthe d-dimensional higher-spin theory, [39], is way more complicated than the 4d one.

The only condition that a spin-tensor has to obey is the symmetry condition — it mustbe symmetric in (each sort of) indices it has. To have all indices symmetrized is muchsimpler than to preserve a more general Young symmetry types. But the major benefitof using spin-tensors rather than tensors is that the corresponding tensors, which can berecovered by contracting all spinor indices with γ-matrices, are automatically traceless.The latter property is of great use for practical calculations. Being available for free inspinorial language it is not granted and causes a real headache in the language of tensorswhich is on top of having the definite Young symmetry type. There are deeper reasons ofcourse as to why spinors are so effective, we just mention those from practical standpoint.

In the next section we establish the dictionary, which is a little bit boring, with thesummary table put at the end.

63

8.1 so(3, 1) ∼ sl(2,C)

The idea behind two-component representation for so(3, 1) Lorentz fields is the identity4 = 2× 2. Particularly, any Lorentz vector va, a = 0, ..., 3 can be encoded by two-by-twomatrix

va ↔(m np q

), (8.2)

The convenient basis for 2 × 2-matrices is given by σaαβ

= (I, σi), where I - is the unit-

matrix and σiαβ

— Pauli matrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

), (8.3)

of these we know, thatσi, σj = 2δijI , Tr σi = 0 . (8.4)

The corresponding va is then a hermitian matrix, v = v†,

va ↔(v0 + v3 v1 − iv2v1 + iv2 v0 − v3

). (8.5)

Let us choose Minkowski metric in the form ηab = (− + ++) and introduce dual setto σ-matrices σaαβ = (I,−σi), then for any va we can define

vαβ = vmσmαβ , (8.6)

such that

vm = −12vαβ σ

mβα . (8.7)

Indeed, −12vαβ σ

aβα = −12vbσbαβ σ

aβα = −12vbTr (σbσ

a) = −14vbTr σb, σa = va. It is

convenient then for each type of indices dotted and undotted to introduce antisymmetricmatrices

ǫαβ = iσ2 =

(0 1−1 0

)= ǫαβ , (8.8)

with its inverse −ǫαβ = (iσ2)−1 = −iσ2, ǫαβ = −ǫβα, ǫαγǫβγ = δαβ and define symplectic

sp(2)-form for raising and lowering indices37

Aα = Aγǫγα , Aα = ǫαβAβ . (8.9)

note the position of indices, which, unlike the case of a symmetric metric, can be changedat the price of an additional sign factor38.

37The appearance of the symplectic structure is not accidental since sl(2) ∼ sp(2) thanks to thefollowing identity valid for 2 × 2 matrices, AT ǫA = detAǫ, which implies that an SL(2) matrix A,detA = 1, is at the same a symplectic matrix AT ǫA = ǫ.

38There are different conventions on how to work with antisymmetric metric tensor. Our conventionmay seem unusual at first sight, but at the end it is the simplest one. For example, raising and thenlowering an index one gets back to the original expression. A convention different from ours produces asign factor, which to us is unnatural and requires much care in computations. We are free to change theposition of uncontracted indices using our convention. More detail on symplectic calculus can be foundin Appendix F.

64

Analogously for ǫαβ. Note, that AαBα = −AαBα. Using the above definitions one

finds, that

σmαα = ǫβαǫβασ

mββ , σααm = ǫαβǫαβσmββ , (8.10)

allowing us to identifyσααm ≡ σαα

m . (8.11)

Matrices σmαα with indices being raised and lowered by ǫαβ and ǫαβ are called Van der

Waerden symbols. Two-component spinors Aα and Aα are called chiral and anti-chiral,correspondingly. Let us note that one should be careful with numerical factors in identi-fying scalar combinations, e.g.

vααvαα = −2vava (8.12)

To proceed we need a simple lemma. Any bi-spinor Aαβ can be decomposed as

Aαβ = A(αβ) +1

2ǫαβAγ

γ . (8.13)

The proof is straightforward, decomposing Aαβ = A(αβ) + A[αβ] and noting, that anyantisymmetric 2×2-matrix A[αβ] ∼ ǫαβ one gets (8.13). Using it we can further prove thefollowing relation for σ-matrices

σmα

ασnβα = ηmnǫαβ + (σmn)αβ , (σmn)αβ = −(σnm)αβ = (σmn)βα =

1

2[σm, σn]αβ .

(8.14)Indeed, consider first symmetric part (mn) of l.h.s.

σmα

ασmβα = −σm

βασm

αα =1

2ǫαβσ

mγ

ασmγα = ηmmǫαβ . (8.15)

For antisymmetric part [ab] one obtains the definition of the r.h.s. The great advantageof the 4d spinor formalism is that it allows one to handle easily traceless Young diagrams.Particularly, let us state an important fact which will be very useful in what follows. Givena traceless Lorentz tensor Ca1b1,a2b2,...,asbs corresponding to a rectangular two-row Youngdiagram, i.e. it is antisymmetric with respect to each pair of indices C..,ab,.. = −C..,ba,.., it istraceless with respect to any indices and it satisfies Young condition — antisymmetrizationof any three gives zero C..,[ab,c],.. = 0, then its spinor counterpart reads

Ca1b1,a2b2,...,asbs → Cα(2s), Cα(2s) , (8.16)

where the repeated indices α(2s) = α1 . . . α2s mean total symmetrization as usual.We will demonstrate this statement with the simplest examples. Let us start with

s = 1, i.e. Cab = −Cba. Its spinor image by definition is Cαα,ββ = Cabσaαασbββ . Using(8.13), one can decompose

Cαα,ββ = Cab

(1

2σaαασbββ +

1

2σaβασbαβ +

1

2ǫαβσaγασb

γβ

). (8.17)

Using now ab-antisymmetry we finally obtain

Cαα,ββ = ǫαβCαβ + ǫαβCαβ , (8.18)

65

where

Cαβ = Cβα =1

2σaαγσbβ

γCab , Cαβ = Cβα =1

2σaγασb

γβC

ab . (8.19)

It is crucial that Cαβ is a complex conjugate of Cαβ. A symmetric rank-two spin-tensorhas 3 complex components, i.e. 6 real, which is exactly the number of components of theMaxwell tensor Fmn = −Fnm. If Cαβ and Cαβ were not conjugated to each other, thecorresponding Fmn would be complex. For the same reason, a tensor is often equivalent toa pair of spin-tensors, which are complex conjugated with the only exception for Cα(s),α(s)

which is self-conjugated. In particular, the spinorial version of xa, which is xαα, is ahermitian matrix, (8.5).

The next less trivial example is Cab,cd whose Young symmetries are those of gravityWeyl tensor, taken again in the antisymmetric basis. Its spinor counterpart can be shownto be

Cααββ,γγδδ = Cab,cdσaαασbββσcγγσdδδ = Cαβγδǫαβǫγδ + Cαβγδǫαβǫγδ , (8.20)

where

Cα(4) =1

4Cab,cdσaα

γσbαγσcαδσdαδ (8.21)

is a totally symmetric spin-tensor. Note, that in deriving this result not only the trace-lessness and antisymmetry of ab and cd have been used, but also the Young condition[ab, c] = 0. On top of that one repeatedly exploits (8.13). Analogously, one shows that

Cα1α1β1β1,...,αsαsβsβs= Cα(s)β(s)ǫα1β1

...ǫαsβs+ c.c. , (8.22)

Cα(2s) =1

2sCa1b1,a2b2,...,asbsσa1α

γ1σb1αγ1 ...σasαγsσbsαγs . (8.23)

Finally, let us conclude this section with spinor representation for Minkowski metric ηab

ηαα,ββ = ηabσaαασ

bββ

= 2ǫαβǫαβ (8.24)

8.2 Dictionary

The dictionary between tensors of the 4d Lorentz algebra so(3, 1) and spin-tensors ofsl(2,C) is summarized in the table below.

If we sum over all spins s = 0, 1, 2, 3, ... then we see that the set of one-forms isisomorphic to the set of zero-forms in a sense that they cover the same space of tensorsin the fiber space — all two-row Young diagrams, each appearing once, see Section 5.6.The main statement of this section is that the same space is isomorphic to the space ofall spin-tensors of even ranks, each appearing once, i.e.∑

s>k

ωa(s−1),b(k) ∼∑

s,i

Ca(s+i),b(s) ∼∑

k+m=evenωα(k),α(m) ∼

∑

k+m=evenCα(k),α(m) . (8.25)

Allowing for spin-tensors of odd ranks brings fermions. The same space can be describedas the space of all functions of two auxiliary two-component variables, yα, yα, since theTaylor coefficients cover the same range of spin-tensors,

f(y, y) =∑

k,m

1

k!m!fα(k),α(m)yα...yα yα...yα . (8.26)

66

so(3, 1) tensor sl(2,C) tensor dimension

• • 1Dirac spinor T α ⊕ T α 4

T a ∼ T αα 4

T a,b ∼ T αα ⊕ T αα 6

T aa ∼ T αα,αα 10

T a(k) ∼ k T α(k),α(k) (k + 1)2

T a(k),b(k) ∼ k T α(2k) ⊕ T α(2k) 2(2k + 1)

T a(k),b(m) ∼mk T α(k+m),α(k−m) ⊕ T α(k−m),α(k+m) 2(k +m+ 1)(k −

m+ 1)

The main advantage of using spin-tensor generating function is that f(y, y) is an un-constrained function (with fermions) or obeys a simple constraint f(−y,−y) = f(y, y)(bosons only), while to describe the same space in terms of vector-like variables we needa number of differential constraints, see Section 5.6 and Appendix E.3, which are difficultto deal with, especially in interactions.

There is an alternative way to describe the same space. One can take just one auxiliaryvariable YA with the index A running over 4 values. We can think of it as a compositeindex A = α, α. Then

f(Y ) =∑

s

1

s!fA(s) YA...YA =

∑

k,m

1

k!m!fα(k),α(m)yα...yα yα...yα (8.27)

This fact has again a group-theoretical meaning, so(3, 2) ∼ sp(4,R), which is usefulto remember of. It is reviewed in Appendix G together with the branching rules forso(3, 2) ↓ so(3, 1). The bosonic projection f(−y,−y) = f(y, y) means f(Y ) = f(−Y ).

9 Free HS fields in AdS4

There is no conceptual problem to extend the unfolded equations for a field of any spinto AdSd. However, the unfolded equations for the individual fields are more complicatedthan for all spins gathered into an infinite multiplet of the higher-spin algebra. Thanksto the effective spinorial language the unfolded equations in AdSd with d = 3, 4, 5 aresimpler. We will focus on the higher-spin theory in AdS4. There are no propagatinghigher-spin fields in d < 4, so the case of AdS4 is the first nontrivial one and historicallythe first example of higher-spin theory.

A good warm up starting point on the way is to consider Klein-Gordon equationin Minkowski space-time, i.e. the spinorial version of (5.66), and then generalize theelaborated machinery to AdS4 and to any spin. A content of this section should be

67

compared with Section 5 as we simply convert our d-dimensional findings into spinorialform for d = 4 and then extend the equations to AdS4, which is a minor modification inthe spinorial notation but a challenge in the vector one. Fortunately, one does not needto apply σ-matrices to every equation in Section 5 reaching an equivalent result entirelyin the language of spin-tensors.

9.1 Massless scalar and HS on Minkowski

We are interested in obtaining first order differential equations of motion for a free masslessscalar that will be equivalent to

C = 0 , = ∂m∂m (9.1)

The equations that we are looking for should be formulated in terms of differential forms.The spinorial avatar of coordinate xm is a bispinor xαα. So, let us introduce differentiald = dxm ∂

∂xm = dxαα ∂∂xαα . The most general equation that one can write down for dC has

the formdC = Cααh

αα ↔ ∂ααC = Cαα , (9.2)

where hαα = dxαα is the vielbein in Cartesian coordinates in Minkowski space-time.Formally, hααm = σαα

m or hααββ

= ǫαβ ǫα

β . Field Cαα is some arbitrary field, the spinor

version of Ca. Proceeding further, one writes down the most general equation for dCαα,c.f. (5.58),

dCαα = Cαα|ββhββ ↔ ∂ββCαα = Cαα|ββ , (9.3)

where we once again have introduced a new field Cαα|ββ, which can be decomposed intotraceless and traceful parts, c.f. (5.57),

Cαα|ββ = Cαβ,αβ + Tαβǫαβ + Tαβǫαβ + Tǫαβǫαβ . (9.4)

On the other hand, from (9.2), (11.147) one has Cαα|ββ = ∂ββ∂ααC. Using then (9.1) onefinds that Tαβ = Tαβ = T = 0 and so, c.f. (5.59),

dCαα = Cαβ,αβhββ (9.5)

Keep going in similar fashion we have the decomposition

dCα(2),α(2) = Cα(2),α(2)|ββhββ =

(Cα(2),α(2)β + ǫαβTα,α(2)β + ǫαβTα(2)β,β + ǫαβǫαβTα,α

)hββ .

Observing that ∂ββCα(2),α(2) = ∂ββ∂ααCαα and using [∂αα, ∂ββ] = 0 we immediately findthat Tα,α(3) = Tα(3),α = Tα,α = 0 and so,

dCα(2),α(2) = Cα(3)β,α(2)βhββ . (9.6)

The chain of the obtained equations is infinite and at each level n reads

dCα(n),α(n) = Cα(n)β,α(n)βhββ , n ≥ 0 . (9.7)

68

Its dynamical content is equivalent to that of Klein-Gordon equation for the lowest com-ponent n = 0, C(x) expressing the higher level fields via derivatives of the physical fieldC, c.f. (5.66),

C(x) = 0 , Cα(n),α(n) = ∂αα . . . ∂αα︸ ︷︷ ︸n

C(x) (9.8)

It is said, that fields Cα(n),α(n) for n ≥ 0 constitute a scalar module. It can be shownthat the obtained set of fields that describe a scalar field is the same for any space-timebackground. The dynamical equations for general background of course will be different,yet their form is known only for (A)dSd and Minkowski, [104].

It is convenient to pack the obtained scalar module into a generating function

C(y, y|x) =∞∑

n=0

1

(n!)2Cα(n),α(n)(y

α)n(yα)n , (9.9)

this leads to the following equation39, c.f. (5.68),

dC(y, y|x)− hαα ∂2

∂yα∂yαC(y, y|x) = 0 , (9.10)

which by construction is consistent with integrability condition d2 = 0. Indeed, we needto check that

0 ≡ ddC(y, y|x) = hαα∂2

∂yα∂yα∧ hββ ∂2

∂yβ∂yβC(y, y|x) (9.11)

The latter is obvious with the help of the following identity

hαα ∧ hββ ≡ 1

2Hαβǫαβ +

1

2H αβǫαβ (9.12)

where Hαβ = Hβα = hαγ ∧ hβγ , idem. for H αβ, and the fact that ξαξβǫαβ ≡ 0 for any

two-component commuting object ξα, even ∂α.The consistency does not require specific for a scalar module grading of generating

function (yα

∂

∂yα− yα ∂

∂yα

)C(y, y|x) = 0 , (9.13)

i.e. it does not rely on the number dotted indices being equal to that of undotted. Thisallows us to consider generating function of the form

C(y, y|x) =∞∑

n,m=0

1

m!n!Cα(m),α(n)(y

α)m(yα)n (9.14)

and impose the condition (9.10) to see if it reproduces anything reasonable. It is conve-nient to present C(y, y) as C =

∑∞2s=0Cs(y, y|x), where

Cs =∞∑

n=0

1

(2s+ n)!n!(Cα(n+2s),α(n)(y

α)n+2s(yα)n + Cα(n),α(n+2s)(yα)n(yα)n+2s) , (9.15)

39Peculiarities of differential calculus when the metric is symplectic, i.e. antisymmetric, which is ourcase of ǫαβ, are discussed in Appendix F.

69

such that (yα

∂

∂yα− yα ∂

∂yα

)Cs(y, y|x) = ±2sCs , (9.16)

Note that the grading operator above commutes with the equations of motion, so (9.10)decomposes into independent subsystems for each s. Using the spinor-tensor dictionaryderived in Section 8 and the field content used in Section 5.2 it is easy to see that (9.10)is the spinor version of the unfolded equations for the zero-forms Ca(s+k),b(s), (5.94).

The module C1/2 describes left and right massless s = 1/2 fields Cα, Cα according toDirac equations

∂βαCβ = 0 , ∂αβC

β = 0 , (9.17)

with all of the rest fields in the module C1/2 expressed via derivatives of the fermionmatter fields Cα and Cα. The s = 1 module is equivalent to Maxwell equations andBianchi identities for (anti)self-dual parts of the Maxwell tensor Cαβ ⊕ Cαβ

∂γβCγα = 0 , ∂βγCα

γ = 0 , (9.18)

which are spinorial versions of (5.82) and look more symmetric than (5.82). For s = 2 wearrive at Bianchi identities for gravity Weyl tensor

∂γβCγα(3) = 0 , ∂βγCα(3)

γ = 0 , (9.19)

which are spinorial versions of (5.83). One reproduces equations for generalized HS cur-vatures Cα(2s), Cα(2s) in this fashion, c.f. (5.84),

∂γβCγα(2s−1) = 0 , ∂βγCα(2s−1)

γ = 0 . (9.20)

Similar analysis can be implemented for AdS4 space-time. We will see that it does notlead to much complication.

9.2 Spinor version of AdS4 background

To write down spinor form of equation (3.54) we need to find spinor counterparts forso(3, 2) generators and connection fields. Recall that Lorentz vector xa is representedby bi-spinor xαα, while antisymmetric Lorentz tensor Tab = −Tba by a pair of symmetricTαβ and Tαβ. Therefore, the spinorial version of AdS4 generators Lab, Pa, (3.40), are

the translations Pαβ and Lorentz generators Lαβ, Lαβ. The representation in terms ofspinor generators gives rise explicitly to the well known isomorphism40 so(3, 2) ∼ sp(4,R).Indeed, gathering Lαβ , Lαβ, Pαβ together into symmetric matrix, A,B, ... = 1, ..., 4,

TAB = TBA =

(Lαβ Pαβ

Pβα Lαβ

)(9.21)

40Understanding of sp(4,R) ∼ so(3, 2) is not required in this section as all formulae will be writtendown using the Lorentz covariant basis of sl(2,C) ∼ so(3, 1). The discussion of sp(4,R) ∼ so(3, 2) canbe found in Appendix G.2.

70

and defining the invariant sp(4,R)-form as

ǫAB = −ǫBA =

(ǫαβ 00 ǫαβ

). (9.22)

the sp(4) commutation relations

[TAB, TCD] = ǫBCTAD + ǫACTBD + ǫBDTAC + ǫADTBC (9.23)

when rewritten in terms of sl(2,C) with A = α, α give41

[Lαα, Lββ ] = ǫαβLαβ , [Lαα, Lββ ] = ǫαβLαβ , (9.24)

[Lαα, Pββ] = ǫαβPαβ , [Lαα, Pββ] = ǫαβPβα , (9.25)

[Pαα, Pββ] = λ2(ǫαβLαβ + ǫαβLαβ) , (9.26)

where for historical reason the cosmological constant Λ was replaced by λ2. Its appearancerequires rescaling of the translation generators Tαα → λPαα. For connection fields we havea,b → αβ, ¯ αβ , ha → hαα. The zero-curvature condition (3.54), specialized to so(3, 2)in terms of spinor connections

Ω =1

2ωαβLαβ + hααPαα +

1

2ωαβLαβ (9.27)

acquire the following form

dhαβ +αγ ∧ hγβ + ¯ α

γ ∧ hαγ = 0 , (9.28)

dαβ +αγ ∧γβ = −λ2hαγ ∧ hβγ , (9.29)

d ¯ αβ + ¯ αγ ∧ ¯ γβ = −λ2hγα ∧ hγβ . (9.30)

Let us define Lorentz-covariant derivative D = d+ as follows

DAαα = dAαα +αβ ∧Aβα + ¯ α

β ∧ Aαβ , (9.31)

where Aαα is any q-form. The upshot is that each index of a spin-tensor is acted upon bythe appropriate half of the spin-connection, undotted indices are rotated with αβ, whiledotted indices with ¯ αβ . From (9.28)-(9.30) then we have

D2Aαα = −λ2Hαβ ∧ Aβα − λ2H α

β ∧Aαβ , (9.32)

where we recall that Hαβ = hαγ ∧ hβγ.Altogether we have four equivalent ways of presenting the background 4d anti-de

Sitter geometry in terms of flat connection Ω, which has 10 components, in accordance

41Four terms, 2 × 2, in the first line are needed to symmetrize over the indices denoted by the sameletter. Two terms are in the second line with no additional terms in the third line. Note that α is totallydifferent from α, so that no symmetrization over α, α is possible without breaking the Lorentz symmetry.

71

with various choices of the base

so(3, 1) : Ω =1

2ωa,bLab + haPa 6 + 4 (9.33)

sl(2,C) : Ω =1

2ωαβLαβ + hααPαα +

1

2ωαβLαβ 3 + 4 + 3 (9.34)

so(3, 2) : Ω =1

2ΩA,BTAB 5× 4/2 (9.35)

sp(4,R) : Ω =1

2ΩABTAB 4× 5/2 (9.36)

with the commutation relations given in (3.40), (9.24), (3.41) and (9.23) in terms ofso(3, 1), sl(2,C), so(3, 2) and sp(4,R) bases, respectively.

Poincare coordinates are useful in applications, the spinorial counterpart of (3.57), is

αα =i

2zdxαα , αα = − i

2zdxαα , hαα =

1

2zλ(−dxαα + iǫααdz) , (9.37)

where the coordinates split into the radial coordinate z and three boundary coordinatesxi packed into a symmetric real bispinor xαβ = xβα. All together z and xαβ combine intohermitian xαα = xαβδβ

α+iǫααz, where iǫαα is one of the Pauli matrices, −σ2. The choice ofcoordinates breaks manifest symmetry down to the boundary Lorentz symmetry so(2, 1)that rotates xαβ and dilatations, x, z → γx, γz. The vierbein hαα gives the expectedmetric tensor

ds2 =1

z2λ2(dz2 + dxidx

i) . (9.38)

9.3 Extension to AdS4

First of all let us extend equations (9.10) written in Cartesian coordinates to any coor-dinate system, where spin-connection may not be trivial. This is done similarly to theway we did in Sections 5, 5.3. One can replace d with the Lorentz-covariant derivativeD = d+ to obtain the following equation in Minkowski space-time

DC(y, y|x)− hαα ∂2

∂yα∂yαC(y, y|x) = 0 , (9.39)

where the fact that D acts on every index, (9.31), can be simulated by, c.f. (5.98),

D = d+αβyα∂β +αβyα∂β , (9.40)

where , ¯ , h obey (9.28)-(9.30) with λ = 0, which entails D2 = 0. Our next goal is tolift (9.39) to AdS4 where λ 6= 0 and the system has to be modified.

Technical difference in deriving equations analogous to (9.7) for a scalar in AdS4 isthat covariant derivatives no longer commute in this case D2 6= 0, (9.32). Consider thecase of massive scalar on AdS4 in some detail, i.e. we aim to rewrite in the unfolded formthe Klein-Gordon equation42

DααDααC = −m2C . (9.41)

42The sign in front of m2 is in accordance with the signature (+ −−−).

72

The analysis is analogous to Minkowski case, we have

DC = Cααhαα , (9.42)

where Cαα is yet unconstrained. We can write then

DCαα = Cαα|ββhββ =

(Cαβ,αβ + Tαβǫαβ + Tαβǫαβ + Tǫαβǫαβ

)hββ .

From Cαα = DααC and (9.41) one finds that Tαβ = Tαβ = 0, while T = 14m2C. Proceeding

further

DCα(2),α(2) = Cα(2),α(2)|ββhββ =

(Cα(2),α(2)β + ǫαβTα,α(2)β + ǫαβTα(2)β,β + ǫαβǫαβTα,α

)hββ .

Using Bianchi identities (9.32) one finds, that Tα(3),α = Tα,α(3) = 0, and Tαα ∼ Cαα. Thispattern persists at higher levels and allows one to write general expression

DCα(n),α(n) = Cα(n)β,α(n)βhββ + fnCα(n−1),α(n−1)hαα , (9.43)

where coefficients fn to be determined from Bianchi identities (9.32). Note that the fieldcontent does not change when we switch on λ, the fn term compensates for DD 6= 0. Onone hand, we have

D2Cα(n),α(n) = −λ2Hαβ Cβα(n−1),α(n) − λ2H α

β Cα(n),βα(n−1) (9.44)

On the other, using

DCα(n)β,α(n)β =Cα(n)βγ,α(n)βγhγγ + fn+1

(Cα(n−1)β,α(n−1)βhαα + Cα(n),α(n)hββ+ (9.45)

+ Cβα(n−1),α(n)hαβ + Cα(n),βα(n−1)hβα

)(9.46)

andDCα(n−1),α(n−1) = Cα(n−1)β,α(n−1)βh

ββ + fn−1Cα(n−2),α(n−2)hαα (9.47)

one arrives at the following recurrent condition

−λ2 = −12fn+1(n+ 2) +

1

2nfn , (9.48)

which can be easily solved by

fn = +λ2 +A

(n+ 1)n, (9.49)

where A is related to the mass term in (9.41) A = 12(m2 − 4λ2). The pure massless case,

when a scalar in question is conformal, corresponds to A = 0 and yields the simplest fn

m2 = −4λ2 . (9.50)

In this case we have the following chain of equations

DCα(n),α(n) = hββCα(n)β,α(n)β + λ2hααCα(n−1),α(n−1) , (9.51)

73

which in terms of generating function (9.9) reduces to

DC(y, y|x) = 0 , (9.52)

D = D − hαα ∂2

∂yα∂yα− λ2hααyαyα . (9.53)

Just as in Minkowski space-time, (9.52) is consistent, i.e. d2 = 0, for any C(y, y|x) nomatter if it has or has not the form of a scalar module (9.9). In generic case eq. (9.52)describes all fields with s ≥ 0 propagating in AdS4 along the lines of Section 9.1. Fors ≥ 1 these read, c.f. (9.20),

DβαCβα(2s−1) = 0 , DαβC

βα(2s−1) = 0 . (9.54)

The obtained equation has a very clear algebraic meaning which we reveal in what follows.Obviously, there is a well-defined flat limit, λ→ 0 that results in (9.39).

9.4 HS gauge potentials

So far massless higher-spin fields have been described in terms of generalized curvaturesanalogous to Maxwell s = 1 tensor and s = 2 Weyl tensor, which is parallel to Section5.5. These along with s = 0 and s = 1/2 matter fields reside in zero-form Weyl moduleC(y, y|x). In this section we add gauge potentials aiming at the spinorial, extended toAdS4, version of (5.92)-(5.97).

The simplest case one can start with is a massless s = 1 field. Its gauge potentialA = Aµdx

µ defines Maxwell tensor which is Cαβ, Cαβ components of the Weyl moduleaccording to

dA = hαγ ∧ hβγCαβ + hγα ∧ hγβCαβ . (9.55)

Gauge potential A possesses a standard gauge invariance

δA = dξ . (9.56)

Consider now massless s = 2 case. To do so one needs to impose Einstein equations andlinearize them around AdS4 background. Imposing Einstein equations is equivalent tostate that Riemann tensor differs from that of AdS4 by Weyl tensor. In other words,we can write down the following equations for s = 2 Lorentz connection ωαβ, ωαβ andvierbein field eαα, which are (4.29)-(4.30) in the language of spinors,

Deαα =deαα + ωαγ ∧ eγα + ωα

γ ∧ eαγ = 0 , (9.57)

dωαβ + ωαγ ∧ ωγβ + λ2eαγ ∧ eβγ = eγδ ∧ eδδCαβγδ , (9.58)

dωαβ + ωαγ ∧ ωβγ + λ2eγ

α ∧ eγβ = eδγ ∧ eδδC αβγδ . (9.59)

This system is exact, like (4.29)-(4.30) is, in the sense that it is valid in the full gravityprovided that Weyl tensor obeys the Bianchi identities. Then we linearize it around AdS4

by replacing ωαβ → αβ + ωαβ, eαα → hαα + eαα, where frame fields and h are ofvacuum AdS4. The Weyl tensor for the empty AdS4 vanishes, so Cα(4) ⊕ C α(4) should

74

be taken of the first order. The zeroth order yields the so(3, 2) zero-curvature equations,(9.28)-(9.30). The resulting linearized equations (linear in ω and e) reduce to

Deαα + ωαγ ∧ hγα + ωα

γ ∧ hαγ = 0 , (9.60)

Dωαβ + λ2(hαγ ∧ eβγ + eαγ ∧ hβγ) = hγδ ∧ hδδCαβγδ , (9.61)

Dωαβ + λ2(hγα ∧ eγβ + eγ

α ∧ hγβ) = hδγ ∧ hδδC αβγδ . (9.62)

Eq. (9.58)-(9.59), which are analogous to (4.35)-(4.36), are consistent provided theBianchi identities for the Weyl tensor hold, therefore, (9.60)-(9.62) are consistent un-der the linearized Bianchi identities for Weyl tensor (9.54). To rewrite (9.60)-(9.62) inthe generating form, we pack fields ωαβ, ωαβ and eαβ into quadratic in y, y polynomial

ωs=2 =1

2ωαβyαyβ +

1

2ωαβ yαyβ + eααyαyα , (9.63)

then system (9.60)-(9.62) arranges into

Dωs=2 = hγα ∧ hγ β∂2

∂yα∂yβCs=2(0, y|x) + hαγ ∧ hβγ

∂2

∂yα∂yβCs=2(y, 0|x) , (9.64)

where (since we are in AdS4 till the very end, the cosmological constant is set to 1, λ = 1)

D = D + hαα(yα∂α + yα∂α) . (9.65)

(9.64) needs to be supplemented with (9.52) in the sector of spin-two. Setting y or y tozero on the r.h.s. of (9.64) is a way to project onto the Weyl tensor.

Again, equation (9.64) is consistent without any reference to the structure of s = 2module (9.63) and so can be generalized for one-form generating function

ω(y, y|x) =∞∑

n,m=0

1

m!n!ωα(m),α(n)(y

α)m(yα)n , (9.66)

Dω = hγα ∧ hγβ∂2

∂yα∂yβC(0, y|x) + hαγ ∧ hβγ

∂2

∂yα∂yβC(y, 0|x) . (9.67)

Eq. (9.67) is called the central on-mass-shell theorem, [41, 88]. It is consistent sinceDD = 0. Since the system is linear it decomposes into an infinite set of subsystems forfields of spin s = 1, 2, 3, .... The following number operator commutes with D and allowsone to single out a subsystem for a particular spin

(yα

∂

∂yα+ yα

∂

∂yα

)ωs(y, y|x) = ±2(s− 1)ωs , (9.68)

In particular, ωα(s−1),α(s−1) component is the spinorial avatar of the spin-s vielbein ea(s−1).The set of fields ωα(s−1),α(s−1), such that n+m = 2(s− 1), are avatars of ωa(s−1),b(k). Thel.h.s. of (9.67) reads

Dωα(n),α(m) + hαγ ∧ ωα(n−1),α(n)γ + hγα ∧ ωα(n)γ,α(m−1) . (9.69)

It is set to zero everywhere except for purely holomorphic and antiholomorphic compo-nents driven by C(y, 0) and C(0, y).

75

Summary. Free higher-spin fields including the scalar can be described uniformly bythe following simple system of equations,

Dω(y, y|x) = hγα ∧ hγβ∂2

∂yα∂yβC(0, y|x) + hαγ ∧ hβγ

∂2

∂yα∂yβC(y, 0|x) (9.70)

δω(y, y|x) = Dǫ(y, y|x) (9.71)

DC(y, y|x) = 0 , (9.72)

where we added the gauge transformation law for ω and

D = D + hαα(yα∂α + yα∂α) (9.73)

D = D − hαα∂α∂α + hααyαyα (9.74)

D = d+αβyα∂β +αβyα∂β (9.75)

These equations are the spinorial version of (5.92)-(5.97) extended to AdS4.It is a beneficial exercise to check that the system above is consistent. First one

observes that D2 = 0 and D2 = 043, so the system is gauge invariant and is consistent upto the terms on the r.h.s. of (9.70). The last thing to show is that these terms do notspoil the consistency.

The system describes fields of spins s = 0, 12, 1, 3

2, 2, 5

2, 3, ..., each in one copy. Trunca-

tion to the bosonic sector can be done by requiring ω and C be even functions of y, y,

ω(y, y) = ω(−y,−y) , C(y, y) = C(−y,−y) . (9.76)

After having defined the higher-spin algebra in the next section we will show that thesepeculiar operators, D, D are automatically generated by representation theory.

Let us conclude with a picture that shows the links between the fields of the unfoldedsystem. (9.70)-(9.72).

10 Higher-spin algebras

So far we have obtained linear in fluctuations unfolded equations that describe free fieldsof all spins, (9.70)-(9.72). At the linearized level fields of different spins are independentand the system simply decomposes into a set of decoupled equations each describing afree field of certain spin. Likewise, the su(N) Yang-Mills theory at zero coupling is just asum of N2− 1 Maxwell actions for noninteracting photons. In fact, there is a Lie algebrathat unifies fields of all spins, [2, 105–107].

As we already know from the general discussion of Section 6 on the unfolded approachone-forms should take values in some Lie algebra. So, the Ωω-type terms44, which onesees in (5.92), (9.70), should arise from the linearization of 1

2[W,W ], whereW takes values

43This formula is a bit of abuse of notation as formally the operator in (9.74) does not even satisfythe chain rule. It is a differentiating operator nonetheless when treated within the corresponding twistedrepresentation.

44Ω is a background, (9.33)-(9.36), i.e. h, , while ω collectively denotes free fields of all spins.

76

Figure 3: A picture showing the position of various fields and their mixing via unfoldedequations. The coordinates are the number of undotted/dotted indices carried by a field.The fields connected by links talk to each other via the unfolded equations (9.70)-(9.72).

undotted

dotted

gauge module, one-forms

Weyl module, zero-forms

eα(s−1),α(s−1)

C α(2s)ωα(2s−2)

Cα(2s)

ωα(2s−2)Chevalley-Eilenberg cocycle

in some possibly infinite-dimensional Lie algebra, which we call the higher-spin algebra,g. To recover Ωω one replaces W with Ω + gω, where we introduced a formal expansionparameter g. The HS algebra g contains the anti-de Sitter algebra so(3, 2) ∼ sp(4,R)as a subalgebra since the graviton must belong to the spectrum and it is described byan so(3, 2)-connection. Flat sp(4,R)-connection Ω describes the vacuum over which theperturbation theory is defined. Let us expand the Yang-Mills field strength for W interms of Ω + gω

R = dW +1

2[W,W ] = dΩ+

1

2[Ω,Ω]

︸ ︷︷ ︸vanishes

(9.28)− (9.30)

+ g(dω + [Ω, ω])︸ ︷︷ ︸first order correction

(9.70)

+ g21

2[ω, ω]

︸ ︷︷ ︸second order

(10.1)

Since sp(4,R) ∼ so(3, 2) is a subalgebra of g, we can decompose g into irreducible sp(4,R)-modules45. Since sl(2,C) ∼ so(3, 1) is a subalgebra of so(3, 2), we can also consider amore fine-grained decomposition of g into sl(2,C) irreducible representation. Dependingon the assumptions about the spectrum of fields contained in W one may find the follow-

45It is a general property of any Lie algebra. Any Lie algebra is first of all a representation of the algebraitself, the adjoint one. For a simple algebra, the adjoint is irreducible. For any given subalgebra, thealgebra as a linear space is a representation again, which is reducible in general. It is this decompositionof the algebra taken as a linear space into a direct sum of irreducible representations of its subalgebrathat we consider.

77

ing components of ω (we remind of (8.26) vs. (8.27))46

spin sl(2,C)-content sp(4,R)-content1 ω ω ∼ •2 ωαα, ωαα, ωαα ωAA ∼3 ωα(4), ωα(3),α, ωα(2),α(2), ωα(1)α(3), ωα(4) ωA(4) ∼s ωα(k),α(m), k +m = 2(s− 1) ωA(2s−2) ∼ 2s− 2

It is a valid question to address if there is a Lie algebra, g, that contains sp(4,R) as asubalgebra and has this spectrum of generators under sp(4,R) or sl(2,C). It does exist!The HS algebra, is the fundamental object in the HS theory. There are many differentways to define it. We present the most useful for practical computations below. Thatthe linearized equations (9.70)-(9.72) take the simplest form when a field of every spinappears once naturally suggest this to be the property of the algebra we are looking for47.

The existence of the algebra is important in the Yang-Mills type of theories. Had itnot existed we would have proven a no-go result that these particular set of fields doesnot admit any interactions at all. This could have happened for a different filed multiplet,e.g., for a finite number of fields with at least one of them having spin greater than two.

The information that the free theory tells us is not enough to recover the algebra im-mediately. What we know is the decomposition of g as a linear space under its subalgebrah = sp(4,R), equivalently we know how W decomposes,

g|h = h⊕⊕

s

Vs , Vs = 2s− 2 , (10.2)

W =1

2(ΩAB + ωAB)TAB +

∑

s 6=2

ωA(2s−2)TA(2s−2) , (10.3)

where we singled out the adjoint representation of h itself. By definition, we know the Liebracket on h, i.e. [h, h] = h, (9.23) or (9.24)-(9.26), and we also know [h, Vs] = Vs whichmanifests Vi being an irreducible representation of h. The missing piece of information is[Vi, Vj] =?. To summarize our knowledge we give the known commutation relations forthe generators

[TAB, TCD] = ǫBCTAD + ǫACTBD + ǫBDTAC + ǫADTBC , (10.4)

[TAB, TC(k)] = ǫBCTAC(k−1) + ǫACTBC(k−1) , k = 2s− 2 . (10.5)

Splitting A = α, α we can get the same relations in sl(2,C) basis.The most general ansatz for the missing commutators that preserves sp(4) decompo-

sition would be

[TA(k), TB(m)] =∑

i

αik,m ǫAB...ǫAB︸ ︷︷ ︸

i

TA(k−i)B(m−i) (10.6)

and we need to look for solutions to the Jacobi identity, which should give options forotherwise free coefficients αi

k,m. Here we assumed that generator of every spin can appear

46A,B, ... = 1, ..., 4 are sp(4) indices. They can be split into a pair of sl(2,C) indices, A = α, α.47This assumption turns out to be valid for bosonic algebra only. If one wishes to add fermions a single

copy of every spin would be not enough.

78

at most once or do not appear at all, so there are no additional ’color’ indices carried byTA(k). One can solve the Jacobi identity and find a unique higher-spin algebra, [2], butwe proceed to the effective realization of this algebra in terms of oscillators.

The last comment is that any element f of the higher-spin algebra g can be expandedas fIeI with the basis vectors being TA(k) = eI

f ∈ g , f =∑

k

fA(k) TA(k) , (10.7)

where fA(k) are totally symmetric tensors, which are the ’coordinates’ in the linear spacespanned by eI . The commutation relations [eI , eJ ] = fK

IJ eK, (10.4)-(10.6), can be rewrit-ten in terms of the coordinates fI = fA(k) as well,

(10.4) : [fABTAB, gCDTCD] = (4fA

C gCD)TAD , (10.8)

(10.5) : [fABTAB, gC(k)TC(k)] = (2kfA

B gBC(k−1))TAC(k−1) , (10.9)

which is a shorter way to encode the commutation relations.

⋆-product realization. Let us consider functions of an auxiliary variable Y A that isan sp(2n) vector. There is nothing special about sp(4) in this section and we can extendit to sp(2n). The indices A,B, ... range over 2n values. The reason we need sp(4) in theend is because sp(4) is the symmetry algebra of AdS4, which becomes relevant in the nextsection only. The Taylor coefficients are symmetric sp(2n)-tensors now

f(Y ) =∑

k

1

k!fA(k)Y

A...Y A, (10.10)

and, as we know, upon splitting A = α, α we get the required spin-tensor content inthe case of sp(4). Truncation to bosons is equivalent to f(Y ) = f(−Y ), c.f. (9.76). So, atleast there is a simple way to pack all the coordinates (10.7) into a generating function.

We want to find some operation on the space of functions f(Y ) in Y that inducesan operation with required properties (10.4)-(10.6) or (10.8)-(10.9) and solves Jacobi.Standard product, i.e. f(Y )g(Y ) is a commutative operation, [f(Y ), g(Y )] ≡ 0 and doesnot even lead to (10.8). We need something less trivial and less local, involving thederivatives in Y .

One more idea, that was realized, [105], after the solution was found in [2], is thatthe Jacobi is solved automatically if the Lie bracket [f, g] comes as a commutator [f, g] =f ⋆ g − g ⋆ f from some associative product

f(Y ) ⋆ (g(Y ) ⋆ h(Y )) = (f(Y ) ⋆ g(Y )) ⋆ h(Y ) , (10.11)

and this is the case for higher-spin algebras. We have seen that the standard product istoo simple. There are two ways to represent f ⋆ g, which is an operation that is linear inits two arguments, f and g, and sends them to some other function of Y ,

f(Y ) ⋆ g(Y ) =

∫f(U)g(V )K(U, V ; Y )d2nUd2nV , (10.12)

f(Y ) ⋆ g(Y ) = K

(∂

∂U, Y,

∂

∂V

)f(U)g(V )

∣∣∣∣U=V=Y

. (10.13)

79

The differential form is perfect when the arguments are polynomials. Note that thedifferential form can be rewritten as

f(Y ) ⋆ g(Y ) = f(Y )K

(←−∂

∂Y, Y,

−→∂

∂Y

)g(Y ) . (10.14)

The integral form is suitable when arguments are more general analytic functions. Theassociativity of the product imposes severe restrictions on the form of the kernels K. Still,there are many solutions, which are equivalent up to a change of the basis eJ → AI

J eI .We present the most convenient one. The solution we choose, which is called the Moyal⋆-product, reads

f(Y ) ⋆ g(Y ) =

∫f(U)g(V ) exp

(i(UA − YA)(V A − Y A)

)d2nUd2nV , (10.15)

f(Y ) ⋆ g(Y ) = f(Y ) exp i(←−∂ Aǫ

AB−→∂ B

)g(Y ) , (10.16)

and there is an equivalent form of the first formula

f(Y ) ⋆ g(Y ) =

∫f(Y + U)g(Y + V )e(iUAV A)d2nUd2nV . (10.17)

The second formula is understood as

f(Y ) ⋆ g(Y ) = f(Y )

(1 + i

←−∂ Aǫ

AB−→∂ B −1

2(←−∂ Aǫ

AB−→∂ B)2 + ...

)g(Y ) . (10.18)

Let us prove that ⋆-product yields an example of a higher-spin algebra with all requiredconditions satisfied. Firstly, with the help of the differential form one sees that

YA ⋆ YB = YAYB + iǫAB (10.19)

and hence the algebra does not give a trivial Lie bracket

[YA, YB]⋆ = YA ∗ YB − YB ∗ YA = 2iǫAB . (10.20)

Moreover, one finds that the commutation relations for TAB, (10.4), (10.8), holds with

TAB = − i2YAYB (10.21)

Indeed, expanding the exp-formula up to the third term one finds

fAATAA ⋆ gBBTBB = −1

4fAAgBBYAYAYBYB + 2fA

C gCB

(i

2YAYB

)− 2fABgAB (10.22)

and then gets

[fAATAA, gBBTBB]⋆ = 4fA

C gCBTAB , (10.23)

80

which is (10.8) and is equivalent to (10.4). Therefore, sp(2n) is a subalgebra48 and corre-sponds to quadratic polynomials in Y . The rest of generators are given by eI = TA(k) =YA...YA up to an unessential numerical prefactor.

Using exp formula one also proves (10.9), (10.5), where the following formulae areuseful

YA ⋆ g(Y ) =(YA + i

−→∂ A

)g(Y ) , f(Y ) ⋆ YA =

(YA − i

−→∂ A

)f(Y ) . (10.24)

Finally, we have a Lie algebra, constructed from an associative one, which obeys (10.4),(10.5) and we can look at (10.6),

fA(k)YA...Y A ⋆ gB(m)Y

B...Y B = (10.25)∑

n

ink!m!

n!(k − n)!(m− n)!(fA(k−n)

C(n)gC(n)B(m−n)

)Y A...Y AY B...Y B . (10.26)

Note, that in the commutator terms with even n survive only. With the same exp formulaone can check that the quadratic Casimir operator of sp(2n) is a fixed number −1

2TAB ⋆

TAB = −14n(2n+ 1).

Let us show how the integral realization of star-product works, since it is this formthat will be very handy in less trivial calculations. Let us also stress, that the differen-tial realization of the star-product strictly speaking is well defined only for polynomials,whereas its integral form acts on much broader space of functions thus being relevant forHS analysis with infinite set of fields.

It is assumed that∫sign includes the numerical factor in order to ensure,

1 ⋆ f(Y ) = f(Y ) ⋆ 1 = f(Y ) , 1 ⋆ 1 = 1 , (10.27)

this is achieved by assuming

1 ⋆ 1 =

∫exp

(iUAV

A)d2nUd2nV =

∫δ2n(U)d2nU = 1 . (10.28)

Let us calculate YA ⋆ f(Y ) (f(Y ) ⋆ YA is analogous)

YA ⋆ f(Y ) =∂

∂pAep

BYB

∣∣∣∣p=0

⋆ f(Y ) (10.29)

∫ep

B(YB+UB)f(Y + V )eiUBV B

=

∫eiUB(V B−ipB)+pBYBf(Y + V ) = (10.30)

=

∫δ(V − ip)epBYBf(Y + V ) = f(Y + ip)ep

BYB , (10.31)

from which one gets (10.24). With the help of either realization one finds the followinguseful formulae

[YA, f(Y )]⋆ = 2i∂Af(Y ) , YA, f(Y )⋆ = 2YAf(Y ) ,

[TAB, f(Y )]⋆ = −i

2[YAYB, f(Y )]⋆ = (YA∂B + YB∂A)f(Y ) , (10.32)

TAB, f(Y )⋆ = −i

2YAYB, f(Y )⋆ = −i(YAYB − ∂B∂A)f(Y ) .

48We have an associative algebra with some embedding of sp(2n) into it. By definition, such an algebrahas to be related to the universal enveloping algebra of sp(2n), see extra Section 12.5.

81

Oscillator realization. Let us mention another realization of the same algebra, whichis useful for Fock-type analysis in practice. One starts with good old quantum mechanicalpairs of operators of coordinate/momentum or creation/annihilation (we ignore ~ and signconventions),

[pk, qj] = iδjk . (10.33)

The pair can be packed into one operator YA = qj , pk. We endow YA since it is anoperator. Then, the commutation relations

[YA, YB] = 2iǫAB , ǫAB =1

2

(0 δjk−δjk 0

)(10.34)

are identical to (10.20) up to a similarity transformation for ǫAB. The difference is that YAare operators and the product is not expected to be commutative from the very beginning,while YA are usual commuting variables endowed in addition to the dot-product withnoncommutative ⋆-product. The two constructions are isomorphic.

The higher-spin algebra is just the algebra of all quantum-mechanical operators one canconstruct with q and p, i.e. it is the algebra of all functions f(Y ) = f(q, p), while ⋆-productrealization is just an effective way to compute the product of operators f(q, p)g(q, p).Keeping in mind that p can be represented in the space of functions of q as pk = i ∂

∂qk

we come to the conclusion that the higher-spin algebra is the algebra of all differentialoperators f(Y ) = f(q, ∂q), which is called the Weyl algebra.

Lorentz covariant base. Consider the sp(4,R) case, i.e. A,B, ... = 1, ..., 4, and A =α, α, then we find TAB split as, c.f. (9.21), (9.24)-(9.26),

TAB ←→ Lαβ = Tαβ, Pαα = Tαα, Lαβ = Tαβ (10.35)

with the help of (10.32) one finds the realization of Lorentz and translation generators bydifferential operators in yα, yα

[Lαβ , f(Y )]⋆ = −i

2[yαyβ, f(Y )]⋆ = (yα∂β + yβ∂α)f(Y ) , (10.36)

[Pαα, f(Y )]⋆ = −i

2[yαyα, f(Y )]⋆ = (yα∂α + yα∂α)f(Y ) . (10.37)

We will need in what follows one more peculiar operator, given by an anticommutator,

Pαα, f(Y )⋆ = −i

2yαyα, f(Y )⋆ = −i(yαyα − ∂α∂α)f(Y ) . (10.38)

Back to the free equations. Now we can see some manifestation of the algebra in thefree equations (9.70)-(9.72), where the operators can be identified as follows

Dω = Dω + hαα(yα∂α + yα∂α)ω = Dω + hαα[Pαα, ω]⋆ , (10.39)

DC = DC − ihαα ∂2

∂yα∂yαC + ihααyαyαC = DC + hααPαα, C⋆ , (10.40)

D = d+αβyα∂β + ¯ αβ yα∂β = d+1

2ωαβ[Lαβ , •]⋆ +

1

2ωαβ[Lαβ, •]⋆ , (10.41)

82

where • is a placeholder to be replaced with the actual expression the operator acts on.The appearance of the anti-commutator in D is crucial and will soon be explained. Theappearance of some i-factors in D as compared to (9.74) can be compensated by rescalingCα(k),α(m) → gn,mC

α(k),α(m), which we did not use in Section 9. This freedom to rescalewas fixed somewhat arbitrary by normalizing the first term on the r.h.s. of (9.43) not tohave any prefactors. The HS algebra tells us that the i-factors are more natural.

Star-product allows us to define AdS frame fields as one-form components to variousbilinears of y and y supplemented with star-product zero-curvature condition. Indeed,using (10.32) one can easily convince oneself, that the following sp(4)-connection

Ω =1

2ΩABTAB =

1

2ωαβLαβ + hααLαα +

1

2ωαβLαβ (10.42)

= − i4(ωαβy

αyβ + ωαβ yαyβ + 2hααy

αyα) (10.43)

substituted intodΩ+ Ω ⋆ Ω = 0 (10.44)

results in (9.28)-(9.30). Then, (10.39)-(10.40) read now49.

Dω = dω + Ω ⋆ ω + ω ⋆ Ω = dω +1

2ΩAB[TAB, ω]⋆ = dω + Ω, ω⋆ , (10.46)

DC = dC + Ω ⋆ C − C ⋆ π(Ω) , (10.47)

where π flips the sign of translations,

π(Pαα) = −Pαα , π(Lαβ) = Lαβ , π(Lαβ) = Lαβ , (10.48)

eventually turning the commutator into anti-commutator inside D in the sector of Pαα.Now the free equations (9.70)-(9.72), which to be viewed as the linearization of a yet

unknown theory, acquire plain algebraic meaning in accordance with general statementsmade about the unfolded equations, see Section 6.

There are two master-fields, the gauge field ω that takes values in the higher-spinalgebra is a usual Yang-Mills connection of that algebra with somewhat different usage.Dω is a linearization of dW +W ⋆W with W = Ω + gω, where Ω is already flat, whichguarantees DD = 0. Another master field is C which also takes values in the higher-spinalgebra, but now the action of the algebra on itself is twisted by π.

49Note that Ω, ω is a mock anti-commutator, it is a commutator since one-forms anti-commute.Indeed,

Ω, ω⋆ = Ωmdxm, ωndxn⋆ = [Ωm, ωn]⋆ dx

m ∧ dxn . (10.45)

This is a difference between the genuine Lie algebra bracket, where, like in Yang-Mills, we have [Ω, ω],and the bracket constructed as a commutator in associative algebra, where [a, b] = a ⋆ b − b ⋆ a. In thelatter case we have to take into account every additional grading, like the differential form degree, whichelements can have, which sometimes turns commutator into anti-commutator. It is easy to translatethe Yang-Mills formulae to the case when the Lie algebra is constructed from an associative one. Forexample, the Yang-Mills curvature is dA+A ⋆A instead of dA+ 1

2 [A,A] and the commutator is implicitbecause one-forms anti-commute. Then δA = dǫ+A ⋆ ǫ− ǫ ⋆ A, etc.

83

Let us comment more on the twisted action. Suppose we have an algebra, say g, and itacts on some linear space, say V , by operators ρ : g→ End(V ), i.e. V is a representationof g. Given any automorphism π of g, i.e. π : g → g and π([a, b]) = [π(a), π(b)], we candefine another action on the same space V , called twisted action, ρπ(a) = ρ(π(a)). Onecan check that it is a representation of g, i.e. ρπ([a, b]) = [ρπ(a), ρπ(b)].

Now, π as defined in (10.48) is an automorphism of the anti-de Sitter algebra, so(3, 2).It acts on P only, so the nontrivial relations to check are (we drop the indices)

[π(P ), π(P )] = [−P,−P ] = [P, P ] = L = π(L) , (10.49)

[π(L), π(P )] = [L,−P ] = −[L, P ] = −P = π(P ) . (10.50)

Formally, π can be realized either as yα → −yα or yα → −yα, since L ∼ yy, L ∼ yy andonly P ∼ yy is affected by such an action,

π(y, y) = (−y, y) , π π = 1 , or π(y, y) = (y,−y) , π π = 1 . (10.51)

With this definition π can be extended to the full higher-spin algebra, g, as an associativealgebra, i.e. π(f ⋆ g) = π(f) ⋆ π(g). Indeed, π just checks if the function is even or oddin y or y and even⋆even=even, odd⋆odd=even, even⋆odd=odd. Hence, it extends to g asa Lie algebra under the commutator.

Finally, the linearized equations for massless fields of all spins combined into a singlemultiplet read as in (9.70)-(9.72) and D, D have the meaning of adjoint and twisted-adjoint covariant derivatives for the master fields ω, C taking values in the higher-spinalgebra50. The nontrivial gluing term on the r.h.s. of (9.70) is the Chevalley-Eilenbergcocycle, see around (6.25) and extra Section 12.4 for more detail. Hence, all terms in theequations have clear representation theory meaning.

’Pure gauge’ solutions. Equation (10.44) is the zero-curvature condition. Hence, anysolution of (10.47) admits pure gauge form

Ω = g−1(Y |x) ⋆ dg(Y |x) . (10.52)

Equation (10.47) is the covariant constancy condition in the twisted-adjoint representationof the HS algebra. The general solution of (10.47) is

C(Y |x) = g−1 ⋆ C0(Y ) ⋆ π(g) , (10.53)

where C0(Y ) is an arbitrary x-independent function. Pure gauge form of (10.52) and(10.53) may look misleading for it seemingly suggests that one can gauge away any solu-tion for dynamical fields. This is not the case. There are two restrictions that constraintgauge functions g(Y |x) in (10.52). First, g(Y |x) should be such that corresponding con-nection Ω be of the form (10.43) i.e. bilinear in y’s and second, it should not provide onewith degenerate vierbein hαα. In practical calculations the gauge function that reproduces

50That gauge fields ω and zero-forms C take values in the same space is a kind of operator-statecorrespondence. π is related to the Chevalley involution, on the CFT side, it is realized as an inversion.

84

vacuum frame fields is some exponent of bilinears in y’s. The fact that higher-spin equa-tions for zero-from C(Y |x) acquire pure gauge representation is a remarkable property ofthe on-shell integrability of this system uncovered with the aid of the unfolding approach.

Let us note, that twisted-adjoint equation (10.47) is fixed by corresponding zero-curvature condition in the HS algebra. It means, in particular, that scalar mass term(9.50) is completely fixed by the HS algebra as well. Unfolded form of dynamical equationsmakes their symmetries manifest. For example, AdS4 global symmetries are governed bythe symmetry parameter ξ(Y |x), collective Killing, that leaves the vacuum, Ω, invariant

0 = δΩ = DΩξ , DΩ = d+ [Ω, •]⋆ , (10.54)

which can be explicitly found once g(Y |x) is known

ξ(Y |x) = g−1 ⋆ ξ0(Y ) ⋆ g . (10.55)

This is a covariant generalization of the Poincare lemma, which shows that DΩξ = 0 anddξ = 0 have isomorphic solution spaces. It is also clear why the matter and HS curvaturesmodule C(Y |x) is described by differential zero-form, rather than some p-form. It makes

it gauge invariant, otherwise there would be gauge invariance δC(p) = Dξ(p−1), where Dis the twisted-adjoint covariant derivative given in (10.40).

One may ask if it is possible to solve for ω in a ’pure gauge’ form. It cannot be justg−1 ⋆ ω0(Y ) ⋆ g since the latter is a solution to the homogeneous equation Dω = 0. Onthe other hand, all gauge-invariant information is in the zero-forms, which contain Weyltensors and derivatives thereof, so it must be possible to reconstruct ω from C up to puregauge solutions Dξ. The formula does exist and can be found in [108].

Klein operator. Let us have a look at the automorphism π, which is one of the keyingredients of the higher-spin theory. Formally, π can be realized as

κ = (−)Ny , κyκ = −y , κyκ = y , (10.56)

where Ny is the number operator yγ∂γ that counts y.Let us find out whether automorphism (10.51) is the inner one or not. In other words

we would like to see if it can be realized in terms of star-product (10.15)-(10.16). Assumingthat such element κ does exist in the star-product algebra such that

κ ∗ f(y, y) ∗ κ = f(−y, y) (10.57)

and κ ⋆ κ = 1 since ππ = 1 one has

κ ∗ yα = −yα ∗ κ , κ ∗ yα = yα ∗ κ . (10.58)

Using (10.32) the second condition in (10.58) tells us that κ is y-independent, while thefirst is equivalent to

κyα = 0 → κ ∼ δ2(y) . (10.59)

From (10.58) it follows also that [κ∗κ, yα]∗ = 0 and therefore κ∗κ ∼ const. The constantcan be chosen to be 1 which leads to

κ = 2πδ2(y) . (10.60)

85

Indeed,

κ ⋆ κ = (2π)2∫du dv δ(y + u)δ(y + v)eiuαvα = 1 . (10.61)

Note, that in checking (10.61) one really has to use the integral form of the star-product,for ⋆-product of δ-functions is out of reach for differential star-product. The operator κwhich satisfies the condition

κ, yα⋆ = 0 , κ ⋆ κ = 1 , (10.62)

is called holomorphic Klein operator. Analogous operator can be defined in the anti-holomorphic sector. As one can see, the Klein operator, being a delta function, strictlyspeaking does not belong to the ⋆-product algebra and hence the automorphism (10.51)is not the inner one. Nevertheless, representation (10.60) is very useful in practice. Forexample the action of Klein operator on a function is just the Fourier transform51 withrespect to holomorphic oscillator y

κ ⋆ f(y, y) =

∫dvf(v, y)eivαy

α

. (10.63)

Automorphism (10.51) in terms of Klein operator action is given simply by

π(f(y, y)) = κ ⋆ f ⋆ κ . (10.64)

Using (10.64) one can derive the Penrose transform that maps solution of field equations(10.47) to AdS4 HS global symmetries. Indeed, having any HS global symmetry param-eter ξ(y, y|x) that satisfies (10.54) one immediately generates solution to twisted-adjointequation (10.47)

C(y, y|x) = ξ(y, y|x) ⋆ κ =

∫ξ(u, y)e−iuαyα . (10.65)

Let us note, that (10.65) can be applied to AdS4 global symmetry parameter which beingbilinear in Y ’s will generate some ill-defined solution via Fourier transform of quadraticpolynomial. Generic HS global symmetry parameter ξ0(Y ) in (10.55), however, can be anarbitrary star-product function which may lead to a well defined Fourier transform.

Let us stress that with the help of κ one can map twisted-adjoint fields in adjoint onesand vice verse. For example,

dC + Ω ⋆ C − C ⋆ κ ⋆ Ω ⋆ κ︸ ︷︷ ︸π(Ω)

= 0 ⇐⇒ d(C ⋆ κ) + Ω ⋆ (C ⋆ κ)− (C ⋆ κ) ⋆ Ω = 0

where the second equation is the first one ⋆-multiplied by κ since dκ = 0 and κ ⋆ κ = 1.The automorphism π may look like an isomorphism between the adjoint and twisted-

adjoint representations. However one has to be careful with this point of view as thefield-theoretical interpretation is totally different. Particularly, the physical domains oftwisted-adjoint and adjoint representations do not overlap much since polynomials aremapped by ⋆κ into derivatives of δ functions and vice versa. More detail on how to workwith ⋆-products can be found in extra Section 12.6.

51The appearance of Fourier transform F should not be surprising since (F F )[f(y)] = f(−y).

86

11 Vasiliev equations

In this section we consider nonlinear interactions for bosonic HS fields governed to allorders by the Vasiliev equations. Firstly, we are going to discuss what one should expecton the way of constructing the interactions within the unfolded approach. Secondly,starting from certain mild assumptions we attempt to derive the Vasiliev equations. Itis not that easy to find reasonings that lead to these assumptions, but apart from thesegaps the derivation is quite solid. Thirdly, we look at the AdS4 vacuum solution toVasiliev equations and show that the linear equations derived in Section 9 do emerge inthe first order perturbative expansion. Then, we discuss certain nontrivial properties ofthe equations like the local Lorentz symmetry being part of the physical requirementsthat should fix the form of the equations. Lastly, we make some comments on higherorders and analyze the Vasiliev system explicitly to the second order.

11.1 Generalities

Up to this point we have shown that the specific multiplet of free HS fields, where a fieldof every spin appears in one copy, can be described in terms of two master-fields ω(Y |x)and C(Y |x), both taking values in the higher-spin algebra, with the empty anti-de Sitterspace given by the flat so(3, 2) ∼ sp(4,R)-connection Ω, (9.28)-(9.30),

dΩ = −Ω ⋆ Ω , (11.1)

dω = −[Ω, ω] + V(Ω,Ω, C) , (11.2)

dC = −Ω ⋆ C + C ⋆ π(Ω) , (11.3)

where there are two crucial ingredients: automorphism π, (10.48), (10.57), and cocyleV(Ω,Ω, C), (9.70). If we believe in the existence of the consistent nonlinear theory thenthese equations must be the linearization of

dω = V(ω, ω) + V(ω, ω, C) + V(ω, ω, C, C) + ... = F ω(ω,C) , (11.4)

dC = V(ω,C) + V(ω,C, C) + V(ω,C, C, C) + ... = FC(ω,C) , (11.5)

where the vertices V(•, ..., •) are linear in each slot and correspond to certain couplings.The first nontrivial couplings V(ω, ω) and V(ω,C) are governed by the higher-spin algebrawith the help of π,

−V(ω, ω) = ω ⋆ ω −V(ω,C) = ω ⋆ C − C ⋆ π(ω) (11.6)

The cocycle V(ω, ω, C) is not known in full, only its value in the AdS vacuum V(Ω,Ω, C)is available. Linearization is carried out by replacing ω → Ω + gω and picking up theterms of the zeroth and first order in g.

To proceed to nonlinear level one can in principle search order by order in perturbationtheory for such a deformation of (11.2) and (11.3) within the set of fields w(y, y|x) andC(y, y|x) that are consistent with the integrability condition d2 = 0 (hence possessinggauge symmetry) and reproduce correct free field dynamics. The perturbative analysis(11.4)-(11.5) is a challenging technical problem and gets already cumbersome at the second

87

order, [40, 41]. This analysis was carried out up to V(ω, ω, C, C), [109], followed by theeffective method of summing all orders, [34, 36, 42, 43].

By looking at the pure gravity case we have to accept the fact that the interactionseries does not truncate to a finite number of terms within the unfolded approach, i.e.all powers of C will appear. Unfortunately, no closed form for F ω and FC is known.Functions F ω and FC are analogous to the expansion of the Riemann tensor in terms ofthe metric field. In gravity the series goes to infinity being though ideologically simple asthe source of infinite tail is the expansion of inverse metric g−1

µν .The Vasiliev equations are the generating equations written in an extended space

that sum up the infinite series into the equations that are no more than quadratic infields. Namely, the equations consist of zero-curvature equation plus certain constraintsthat determine the embedding of interaction vertices into the fields obeying the zero-curvature condition. The interaction vertices V(•, ..., •) can be obtained by solving theequations order by order. Vasiliev equations do not give (11.4)-(11.5) immediately! Theyare formulated in certain extended space with more coordinates and (11.4)-(11.5) appearas solutions in the perturbative expansion around the AdS4 vacuum. In some senseVasiliev equations are analogous to simple differential equations encoding complicatedspecial functions, but in the Vasiliev case the equations encode complicated equations assolutions to additional variables. The last remark is that Vasiliev equations share manyproperties with integrable equations. Integrability usually means that a theory can beput into the form of some covariant constancy or zero-curvature equation, i.e. equationsthat are no more than quadratic in fields.

As an example we will derive second order corrections in the formal coupling constantg, ω → Ω+ gω1 + g2ω2 from Vasiliev system

Dω2 = V(ω1, ω1) + V(Ω,Ω, C2) + V(Ω, ω1, C1) + V(Ω,Ω, C1, C1) (11.7)

DC2 = V(ω1, C1) + V(Ω, C1, C1) (11.8)

and we will show some of these vertices explicitly. And at the order N one can write

DωN =′∑

n+m=N

V(ωn, ωm) +′∑

n+m+k=N

V(ωn, ωm, Ck) + V(Ω,Ω, CN ) (11.9)

DCN =′∑

n+m=N

V(ωn, Cm) (11.10)

where the primed sum∑′ designates that the background covariant derivative has already

been extracted. For future use let us note that at order N the fields of order N appearthe same way as ω1 and C1 appear in the linearized equations. So the equations for ωN

and CN look like the linear equations plus sources built of the lower order fields.The perturbation theory in certain auxiliary variables for Vasiliev equations yields

the perturbation theory (11.9)-(11.10) for (11.4)-(11.5), from which the vertices V can berecovered. It is, of course, makes more sense to look for exact solutions and try to givecertain physical meaning to the extended space.

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11.2 Quasi derivation of Vasiliev equations

It is a sort of conventional wisdom that Vasiliev equations cannot be derived rather oneis welcome to check that they do satisfy all the required assumptions. Below we attemptto put things in a perspective of ’derivation’ starting from the main assumption that theyshould have ’almost’ zero-curvature/covariant constancy form as in integrable theories52,but the field content and the space where the equations are formulated can possibly beextended.

Let W be the full master-field on possibly extended space, of which ω(Y |x) is asubspace. There must be W ⋆ W-term on the r.h.s. of (11.4) where ⋆ as well actson possibly extended space and reduces to the usual HS algebra ⋆ for ω ∈ W. Denotingall other corrections on the r.h.s. of (11.4) as Φ, which must be a two-form, and usingd2 = 0 one finds

dW +W ⋆W = Φ , (11.11)

dΦ+ [W,Φ]⋆ = 0 , (11.12)

where the second equations comes from consistency requirement. Unfolding tells us thatthis system has local gauge invariance53

δW = dǫ+ [W, ǫ]⋆ + ξ1 , δΦ = dξ1 + W, ξ1⋆ − [ǫ,Φ]⋆ . (11.13)

Suppose Φ is a fundamental field, i.e. is not expressed in terms of some other degree-zeroor degree-one fields. Then its has its own one-form gauge parameter ξ1, which is able tokill W. Therefore, Φ cannot be fundamental.

From free level analysis we have found an appropriate set of fields for the descriptionof gauge field dynamics. These are the master fields — a one-form ω(Y |x) and a zero-formC(Y |x). Equation (11.2) hints to identify somehow Φ with C (for a moment we forgetabout issues with the twisted-adjoint representation). The problem is that Φ is a two-formand hence one needs some extra two-form to identify Φ ∼ C. The only field independenttwo-form that one has is dxa ∧ dxb. This, however, carrying indices should be contractedwith some derivatives in Y ’s of C, thus bringing us to annoying perturbative analysis.

A way out proposed by Vasiliev is to introduce some auxiliary direction to space-timethat will allow one to determine the two-form carrying no indices. This can be done oneway or another, but this additional direction can be anticipated to be auxiliary for Yas well. So, in spinorial formalism we are working with, let us choose some auxiliary54

ZA = (zα, zα) and corresponding differential dZA, which anticommutes with dxm andprovides us with the differential two-forms. If the auxiliary space is two-dimensional thetwo-form is a top-form and is unique, in particular there are no free indices it can carry,

52For example, the famous Korteweg-de Vries equation, which contains higher-derivative and is non-linear — oversimplified model of higher-spin theory, can be formulated as a zero-curvature equation incertain extended space. Upon solving for the extra functions certain first-order equations coming asthe components of the zero-curvature condition one gets the KdV equation. Vasiliev equations functionanalogously.

53Mind that [•, •] sometimes changes to •, • due to differential form degree.54We used the same notation, α, for the auxiliary indices to anticipate that z are in close interrelation

with y.

89

dzα ∧ dzα and dzα ∧ dzα. That we need two auxiliary two-forms is due to the two termson the r.h.s. of (9.70), i.e. due to the fact that the spin-s Weyl tensor Ca(s),b(s) splits intotwo complex-conjugate Cα(2s), C α(2s) that yield two sources for one-forms ω(Y |x). Thisallows us to write down system (11.11) in the extended space by introducing the extendedconnection

W = Wmdxm + Aαdz

α + Aαdzα , (11.14)

where the fields now depend on a new variable ZA as well

ω(Y |x)→W (Y, Z|x) , C(Y |x)→ B(Y, Z|x) , AA = AA(Y, Z|x) . (11.15)

The exterior differential gets enhanced too and we label it with the hat

d→ d = dx ⊕ dz ⊕ dz . (11.16)

As a result, one may propose the following system

dW +W ⋆W = Φ + Φ , (11.17)

dΦ + [W,Φ]⋆ = 0 , (11.18)

dΦ + [W, Φ]⋆ = 0 , (11.19)

where we have implied the following identification

Φ =i

4dzα ∧ dzαB , Φ =

i

4dzα ∧ dzαB , (11.20)

The system is formally consistent and admits local gauge invariance

δW = Dǫ ≡ dǫ+ [W, ǫ]⋆ , δΦ = [Φ, ǫ]⋆ . (11.21)

Let us stress once again that eqs. (11.18) and (11.19) arise from (11.17) as consistencyconditions d2 = 0 and are not independent. So far nothing is said about extra spinorvariables zα and zα. Let us choose these to be dual to YA i.e.,

[ZA, ZB]⋆ = −2iǫAB , [ZA, YB]⋆ = 0 . (11.22)

It requires the corresponding extension of the star-product, which we will define a bitlater. Recall now, that HS master field zero-form C(Y |x) is subject to twisted-adjointflatness condition rather than the adjoint one at free level. Therefore, at full level weneed to redefine B(Y, Z|x) using the appropriate Klein operator. Analogously to the freetheory consideration, the explicit form of the Klein operator κ depends on the form of theautomorphism through its star-product realization. At nonlinear level, the twisted-adjointautomorphism is defined as

π(yα, yα, zα, zα) = (−yα, yα,−zα, zα) , (11.23)

π(yα, yα, zα, zα) = (yα,−yα, zα,−zα) . (11.24)

The corresponding to be found Klein operators κ and κ are determined by the conditions

κ ⋆ F (Y, Z) = F (π(Y, Z)) ⋆ κ , κ ⋆ F (Y, Z) = F (π(Y, Z)) ⋆ κ , (11.25)

90

where F (Y, Z) is an arbitrary function. Just as in the free case, the twisted-adjointHS-curvature zero-form B(Y, Z|x) is reproduced from the adjoint one as

B → B ⋆ κ , B → B ⋆ κ . (11.26)

Let us note that the fields B and B are not independent as might seem from (11.20)which is in accordance with the linearized description where we had a single Weyl moduleC(y, y|x). This fact imposes severe restriction on the form of higher-spin interactions.Indeed if B and B were independent, equations (11.17) would be just a definition ofthe curvature two-form in dzα ∧ dzα and dzα ∧ dzα sectors on its right hand side. Theexplicit form of Klein operators can be derived not until the extended (Y, Z) star-productis defined, so let us proceed with its definition.

11.3 Extended star-product, Klein operators, HS equations

Commutation relations (11.22) can be reached via the following star-product

f(Y, Z) ⋆ g(Y, Z) =

∫dUdV f(Y + U,Z + U)g(Y + V, Z − V )eiUAV A

. (11.27)

Mind the minus sign in the second argument of g-function which guarantees that [Y, Z]⋆ =0. Star-product is again can be shown to be associative. It reduces exactly to the Moyalstar-product (10.17) once functions f and g are Z-independent. On the space of Y -independent functions we find a formula similar to (10.17) with an extra minus sign inthe exponent that ensures (11.22.a). Commutation relations (11.22) then can be easilyreproduced from definition (11.27). The following simple formulas will be useful for star-product calculations

epAYA ⋆ f(Y, Z) = ep

AYAf(Y + ip, Z − ip) , f(Y, Z) ⋆ epAYA = ep

AYAf(Y − ip, Z − ip) ,ep

AZA ⋆ f(Y, Z) = epAZAf(Y + ip, Z − ip) , f(Y, Z) ⋆ ep

AZA = epAZAf(Y + ip, Z + ip) .

Particularly, from these relations it follows, c.f. (10.24),

YA ⋆ f =(YA + i

∂

∂Y A− i ∂

∂ZA

)f , f ⋆ YA =

(YA − i

∂

∂Y A− i ∂

∂ZA

)f , (11.28)

ZA ⋆ f =(ZA + i

∂

∂Y A− i ∂

∂ZA

)f , f ⋆ ZA =

(ZA + i

∂

∂Y A+ i

∂

∂ZA

)f . (11.29)

It is easy to check that thus defined star-product is associative and provides normalordering for a±A = YA ± ZA, indeed, as follows from (11.28), (11.29)

f ⋆ a+ = fa+ , a− ⋆ f = a−f . (11.30)

The following simple formulas we find useful below, c.f. (10.32),

[YA, f ]⋆ = 2i∂

∂Y Af , [ZA, f ]⋆ = −2i

∂

∂ZAf , (11.31)

YA, f⋆ = 2(YA − i

∂

∂ZA

)f , ZA, f⋆ = 2

(ZA + i

∂

∂Y A

)f , (11.32)

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[− i2YAYB, f ]⋆ =

(YB − i

∂

∂ZB

) ∂

∂Y Af +

(YA − i

∂

∂ZA

) ∂

∂Y Bf , (11.33)

− i2YAYB, f⋆ = −i

(YA − i

∂

∂ZA

)(YB − i

∂

∂ZB

)f + i

∂

∂Y A

∂

∂Y Bf . (11.34)

Klein operators can now be easily derived from their definition (11.25). We alreadyknow from (10.62) that in case of Z-independence the Klein operator is a delta-function of(anti)holomorphic oscillator. Similar consideration for Y -independent functions leads todelta-functions κz = 2πδ2(z) and κz = 2πδ2(z). As a result, the complete Klein operatorsthat satisfy (11.25) are given by

κ = κy ⋆ κz = (2π)2δ2(y) ⋆ δ2(z) = eizαyα

, (11.35)

κ = κy ⋆ κz = (2π)2δ2(y) ⋆ δ2(z) = eizαyα

. (11.36)

Note, that the Klein operators turned out to be regular functions in the extended star-product algebra and therefore automorphism (11.23), (11.24) is the inner one. One cancheck the following straightforward properties

κ ⋆ κ = 1 , κ ⋆ f(y, z) = f(z, y)eizαyα

, (11.37)

κ ⋆ f(y, z) = f(−y,−z) ⋆ κ . (11.38)

analogously for antiholomorphic Klein κ. Note the interchange of variables y and z withinthe arguments of f(y, z). The equations that account properly for the twist automorphismcan be now obtained via substitution (11.26) and (11.35), (11.36) into (11.17)-(11.19)

dW +W ⋆W = Φ ⋆ κ + Φ ⋆ κ , (11.39)

dΦ +W ⋆ Φ− Φ ⋆ π(W) = 0 , Φ =i

4dzα ∧ dzαB , (11.40)

dΦ +W ⋆ Φ− Φ ⋆ π(W) = 0 , Φ =i

4dzα ∧ dzαB . (11.41)

The first equation tells us that curvature two-from (11.39) is only allowed to be nonzeroin auxiliary dZ ∧ dZ sector being pure gauge in space-time. This is a generic feature ofthe unfolded equations that tend to get rid of space-time dependence and reformulatethe dynamics in the auxiliary “twistor” space. Using definition (11.14) let us rewrite theabove equations in the component form.

Before doing this we make one comment. In obtaining (11.39)-(11.41) we carried outfield redefinition Φ→ Φ ⋆ κ, Φ→ Φ ⋆ κ to meet the twisted-adjoint requirement. Whileit is fine to do so in (11.17), the substitution into (11.18) and (11.19) seemingly producesterms of the form ∂z(B ⋆ κ) and ∂z(B ⋆ κ) which do not allow one to drag the Kleiniansthrough the Z–derivative because of their Z–dependence. It does not happen though asthese terms simply do not show up. Indeed the three-forms appearing on l.h.s. of (11.40)and (11.41) are identically zero for dz ∧ dz ∧ dz ≡ 0 and dz ∧ dz ∧ dz ≡ 0. Another typeof potentially dangerous terms ∂z(B ⋆κ) and ∂z(B ⋆ κ) are harmless since ∂z(κ) = 0 and∂z(κ) = 0.

The fact that there are no integrability conditions in zzz and zzz sectors for dimen-sional reason (spinorial indices take two values) suggests that it might be possible to

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impose some extra constraints consistent with equations (11.39)-(11.41). It turns out thisis what one actually should do to describe irreducible nonlinear equations for HS bosonicfields for d = 3 [110] and arbitrary d [39] systems. In other words, systems (11.17)-(11.19)typically have some spurios solutions due to lack of constraint in extra twistor space.Indeed the way it is written in (11.17)-(11.19) the system just tells us in which sector theHS curvature is allowed to be non-zero, (11.17), without specifying the curvature itself— since the rest conditions (11.18) and (11.19) are simply the integrability consequences.The four dimensional case is to some extent peculiar for it was already pointed out thatfields Φ and Φ are not independent rather related to each other through the Klein oper-ators. This fact stipulates some restriction on the possible form of higher-spin curvaturethat enters r.h.s. of (11.17) and takes place only in the case of four dimensional spino-rial system where the two types of spinor fields — holomorphic and antiholomorphic areavailable. Eventually, the 4d extra constraint will turn out to be equivalent to kinematiccondition for the system (11.39)-(11.41) to be bosonic. The required kinematic conditionis not yet fully there, but it is not going to be a problem to identify it. Impressive is thefact that the desired 4d kinematic condition will provide us with some nontrivial algebraicconstraint which is typical for all available nonlinear HS systems.

What is more, that two-form Φ is expressed in terms of zero-form B makes Φ acomposite field, which solves the problem of the extra gauge symmetry, ξ1, (11.13) thatwould be associated with Φ if it were a fundamental field. But for now let us proceedwith component form of (11.39)-(11.41) which reads

dW +W ⋆W = 0 , (11.42)

dB +W ⋆ B − B ⋆ π(W ) = 0 , dB +W ⋆ B − B ⋆ π(W ) = 0 , (11.43)

∂Aα

∂zα− ∂Aα

∂zα+ [Aα, Aα]⋆ = 0 , (11.44)

dAα + [W,Aα]⋆ −∂W

∂zα= 0 , dAα + [W, Aα]⋆ −

∂W

∂zα= 0 , (11.45)

∂Aα

∂zα+ Aα ⋆ A

α =i

2B ⋆ κ ,

∂Aα

∂zα+ Aα ⋆ A

α =i

2B ⋆ κ , (11.46)

∂B

∂zα+ Aα ⋆ B − B ⋆ π(Aα) = 0 ,

∂B

∂zα+ Aα ⋆ B − B ⋆ π(Aα) = 0 , (11.47)

which results from (11.42)-(11.44) as coefficients of dx∧dx, (11.42); dx∧dz∧dz, (11.43.a),dx ∧ dz ∧ dz, (11.43.b); dz ∧ dz, (11.44); dx ∧ dz, (11.45.a), dx ∧ dz, (11.45.b); dz ∧dz, (11.46.a), dz ∧ dz, (11.46.b); dz ∧ dz ∧ z, (11.47.a), dz ∧ dz ∧ z, (11.47.b). Whilesystem (11.11)-(11.12) is manifestly integrable, it is not so obvious for the componentform (11.42)-(11.47).

One may argue that the form of equations contradicts the perturbative scheme laidout in (11.4)-(11.5) as (11.42) seemingly contains no higher-spin corrections that appearon the r.h.s. of (11.4) already at free level, (11.2). Recall, however, that master fields Wand B having got an extra dependence on extra Z-variable would provide the desired HScorrections for dynamical fields w(Y |x) and C(Y |x) through star-product (11.27). Let usnow enlist some obvious properties of the obtained equations

• The equations are to be bosonic. Note, that there is π-automorphism that enters(11.43). There is however similar equation arising from (11.41) with π replaced by

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π. Therefore the system makes sense only for π(W ) = π(W ), or equivalently

W (−Y,−Z|x) =W (Y, Z|x) ⇐⇒ κ ⋆ κ ⋆ W = W ⋆ κ ⋆ κ . (11.48)

The parity property for master field W (Y, Z|x) is a manifestation of the bosonicnature of the system55. It would imply ω(Y |x) = ω(−Y |x) for the physical field.

Correspondingly, the gauge symmetry parameter ǫ must be an even function tooǫ(Y, Z|x) = ǫ(−Y,−Z|x). It has an immediate consequence

AA(Y, Z|x) = −AA(−Y,−Z|x) ⇐⇒ κ ⋆ κ ⋆ AA = −AA ⋆ κ ⋆ κ , (11.49)

since W and AA = (Aα, Aα) are parts of the same connection. Fields and gaugeparameters take values in the same space, now it is the space of even functionsǫ(Y, Z). That the gauge transformation for A, δAA = ∂

∂ZA ǫ+..., contains a derivativealong Z direction changes the parity of AA as compared to W for which the gaugetransformation δW = dǫ+ ... does not affect the parity in Y, Z space.

The bosonic projection (11.49) immediately implies from (11.45) the correspondingprojection for the zero-forms56

B(Y, Z|x) = B(−Y,−Z|x) ⇐⇒ κ ⋆ κ ⋆ B = B ⋆ κ ⋆ κ . (11.50)

Let us stress that unlike (11.48), condition (11.49) and its consequence (11.50) donot follow from the equations. It is this missing kinematic condition (11.49) thatwill be equivalent to some extra algebraic constraint consistent with (11.42)-(11.47).

• Gauge symmetry. Component form of gauge transformations results from (11.21)upon redefinition (11.26)

δW = dǫ+ [W, ǫ] , (11.51)

δAA =∂ǫ

∂ZA+ [AA, ǫ] , (11.52)

δB = B ⋆ π(ǫ)− ǫ ⋆ B (11.53)

• Purely gauge space-time dependence. From (11.42), (11.43) and (11.45) one canalways determine space-time dependence of the master fields in a pure gauge fashion

W = g−1 ⋆ dg , (11.54)

B = g−1 ⋆ B0(Y, Z) ⋆ π(g) , (11.55)

AA = g−1 ⋆∂g

∂ZA+ g−1 ⋆ A0

A(Y, Z) ⋆ g , (11.56)

where g = g(Y, Z|x) is an arbitrary even function. This is a general statement thatany covariant constancy/zero curvature equations can be at least formally solved in

55It is possible to include fermions into nonlinear system by introducing some extra Klein operators [38].The resulting equations are supersymmetric and contain two copies of each spin.

56When no bosonic constraints imposed one has a strange theory containing bosons described at thefree level by gauge fields and fermions described by gauge-invariant generalized Weyl tensors.

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the pure gauge form. Note that the curvature for Aα, Aα is not entirely zero, whichexplains the extra term in the last line.

These formulas suggest, for example, that one can gauge away W–field that is sup-posed to encode HS gauge potentials. While formally it looks as really the case,do not forget that such “gauging” when applied washes away AdS space-time it-self making its frame field and Lorentz connection equal to zero, see discussionafter (10.52). It raises an important question of admissible gauge transformations,which draw a line between small gauge transformations, which are true gauge trans-formations, and large gauge transformations that relate physically distinguishablesolutions. Space-time independent functions B0 and A0 play a role of initial dataimposed at a given space-time point x0 where g(Y, Z|x0) = 1.

• It is now obvious that F ω(ω,C) and FC(ω,C) in (11.4)-(11.5) are more constrainedthan just by d2 = 0, see discussion around (6.35)-(6.36). Indeed dW +W ⋆W = 0yields dB +W ⋆ B − B ⋆ π(W ) = 0 upon applying B ⋆ κ δ

δWto the former. This

proves (6.35)-(6.36) to all orders.

As it was mentioned already, the system we are dealing with requires the bosonic kinematicconstraint (11.49) which is not a consequence of the equations. Our goal is to rewrite(11.49) in some algebraic way and add it to the system so as to make the bosonic natureof the equations intrinsic and manifest. To proceed in this direction let us first performsome harmless field redefinition with A-field

Aα =i

2(Sα − zα) , Aα =

i

2(Sα − zα) , (11.57)

where S is some new field. The shift of vacuum value of A-field is designed to eliminatepartial derivatives with respect to z-variable in (11.44)-(11.47), the coefficient i/2 is chosenappropriately to account for interplay between [z, f ]⋆ = −2i∂zf and ∂zf terms. Shift(11.57) yields

[Sα, Sα]⋆ = 0 , (11.58)

[Sα, Sβ]⋆ = −2iǫαβ(1 + B ⋆ κ) , [Sα, Sβ]⋆ = −2iǫαβ(1 +B ⋆ κ) , (11.59)

[Sα, B ⋆ κ]⋆ = 0 , [Sα, B ⋆ κ]⋆ = 0 , (11.60)

Note, that (11.60) is a consequence of (11.58) and (11.59). Clearly, should one hadB ⋆ κ = B ⋆ κ = 0 the above commutation relations would simply correspond to twocopies of Weyl algebra generated by Sα and Sα. If B is not equal to zero we see, thatB ⋆ κ is a central element for Sα and B ⋆ κ — for Sα. From (11.60) and (11.49) itimmediately follows

Sα, B ⋆ κ⋆ = 0 , Sα, B ⋆ κ⋆ = 0 . (11.61)

Algebraic condition (11.61) eventually brings us to explicitly bosonic and complete non-linear HS equations. That (11.61) respects the Jacobi identities deserves special attention.Set of equations (11.58)-(11.60) supplemented with (11.61) represents two copies of what

95

is known as deformed oscillators. Consider one copy that can be defined as follows. Letthe generating elements yα and K satisfy the relations

[yα, yβ] = −2iǫαβ(1 +K) , yα, K = 0 . (11.62)

The deformed oscillators (11.62) were originally discovered by Wigner in [111] and hap-pened to be related and Calogero model [112]. It is interesting that if one looks at theJacobi [[yα, yβ], yγ] + cycle = 0 it is not going to hold in general unless indices of yα taketwo values so that antisymmetrization of any three gives identically zero. This is the casefor two copies of deformed oscillators generated by HS master fields (11.58)-(11.60) and(11.61). Another very important property is that the deformation of Heisenberg algebra(11.62) respects sp(2) symmetry. Indeed it is easy to see that generators Tαβ = i

4yα, yβ

form sp(2) algebra[Tαβ , Tγδ] = ǫαγTβδ + (α↔ β) + (γ ↔ δ) (11.63)

which yet rotates yα as a vector

[Tαβ , yγ] = ǫαγ yβ + ǫβγ yα . (11.64)

Deformed oscillator properties (11.63) and (11.64) will be of utter importance in identify-ing local Lorentz symmetry for HS system which guarantees tensor interpretation of thedynamical fields.

Remarkably, it is the deformed oscillator constraint (11.61) that one can additionallyimpose to (11.39)-(11.41) leads to the nonlinear system for bosonic massless fields whichwe can finally write down in the form

dW +W ⋆W = 0 , (11.65)

dB +W ⋆ B − B ⋆ π(W ) = 0 , (11.66)

dSα + [W,Sα]⋆ = 0 , dSα + [W, Sα]⋆ = 0 , (11.67)

[Sα, Sβ]⋆ = −2iǫαβ(1 + B ⋆ κ) , [Sα, Sβ]⋆ = −2iǫαβ(1 +B ⋆ κ) , (11.68)

Sα, B ⋆ κ⋆ = 0 , Sα, B ⋆ κ⋆ = 0 , (11.69)

[Sα, Sα]⋆ = 0 . (11.70)

The system of equations (11.65)-(11.70) is known as Vasiliev nonlinear equations for HSbosonic fields in four dimensions. It has a form of (11.42)-(11.47) upon field redefinition(11.57) with an extra constraint (11.69) that makes this system explicitly bosonic. Writtenthis way it contains some flatness conditions in space-time (11.65)-(11.67) and a set ofalgebraic constraints (11.68)-(11.70) which are nothing but a direct sum of two deformedoscillators given by Sα and Sα. The system correspondingly inherits local gauge invariance

δW = dξ + [W, ξ]⋆ , (11.71)

δB = B ⋆ π(ξ)− ξ ⋆ B ⇐⇒ δ(B ⋆ κ) = [B ⋆ κ, ξ]⋆ , (11.72)

δSα = [Sα, ξ]⋆ , (11.73)

δSα = [Sα, ξ]⋆ . (11.74)

As it was already mentioned, introducing the deformed oscillator anticommutator con-dition (11.69) one imposes extra kinematic constraints (11.49), (11.50) and vice versa.

96

Eq. (11.49) can be easily obtained using e.g., [Sα, B ⋆ κ]⋆ = 0 and Sα, B ⋆ κ⋆ = 0.Once it is proved, eq. (11.50) follows immediately from the fact that S ⋆ S is an evenfunction. Together (11.48) and (11.49), (11.50) consistently imply that the system underconsideration is bosonic.

One thing that was missed so far is the reality conditions for master fields. These aredictated by the star-product properties and free level analysis. Without going into detailswe give the final result

y†α = yα , z†α = −zα , (11.75)

W † = −W , (11.76)

S†α = −Sα , (11.77)

B† = π(B) . (11.78)

An important difference between the free and interacting equations is the doublingof oscillators Y → (Y, Z) and appearance of extra S-field. A question is whether thisprocedure preserves physical degrees of freedom which as we have seen are encoded in asingle function C(Y |x) and whether the linearized approximation results in (9.70)-(9.72).We will see, that while from (11.68) one expresses B in terms of S ⋆ S, perturbativelyit is the S-field that appears to be totally auxiliary and is expressed on-shell in terms ofB-field modulo gauge ambiguity. As for an extra Z-dependence, it turns out to be fixedagain up to a gauge ambiguity by the extra condition (11.69).

Generalizations and reductions. We can ask ourselves to what extent the form ofthe equations (11.65)-(11.70) unique and what the possible generalizations or ambiguitiesin higher-spin interactions are. One assumption that was used in writing down system(11.39)-(11.41) that there are no mixing terms dzα ∧ dzα in the auxiliary space curvaturesector. These terms when present violate local Lorentz symmetry since to convert spinorindices one has to introduce some field Hαα(B) that breaks down the symmetry explicitly.So, this possibility is forbidden by the equivalence principle. Another possible modificationwhich does not ruin formal integrability of the Vasiliev equations is to change B ⋆ κ andB ⋆ κ on the r.h.s. of (11.68) to f⋆(B ⋆ κ) and f⋆(B ⋆ κ), correspondingly. While itis possible to make such a modification one can partially eliminate its effect by fieldredefinition B → F (B) which leaves nontrivial only the phase in complex function f . Inother words the ambiguity one cannot get rid of by field redefinition is given by arbitraryreal function φ(B) that enter (11.68) as exp⋆(iφ⋆(B ⋆κ)) ⋆B ⋆κ in holomorphic part andexp⋆(−iφ⋆(B ⋆ κ)) ⋆ B ⋆ κ in the antiholomorphic. The cases of φ = 0, π/2 correspondsto A and B models correspondingly. Finally, there is a way to introduce fermions in thesystem by doubling the set of fields and add some nondynamical moduli that play the roleof different parameters of interaction, see [38]. It is also possible to truncate the bosonicsystem to the fields with even spins only, s = 0, 2, 4, 6, ....

11.4 Perturbation theory

In perturbation theory one starts with appropriate exact vacuum solution. For a fieldtheory, a classical vacuum is some background having no fields propagating on it. In

97

case of HS gauge theory, the proper vacuum is the AdS space-time as we already know,therefore the vacuum W0 one-form should be taken from (10.43) such that (10.44) issatisfied and as long as no dynamical fields are around we take B = 0 and AA = 0. Onehas then the following exact solution of (11.65)-(11.70)

W0 = Ω = − i4(ωαβy

αyβ + ωαβ yαyβ + 2hααy

αyα) =1

2TABΩAB , (11.79)

B0 = 0 , (11.80)

S0α = zα , S0α = zα , (11.81)

where Ω is a good old AdS4 flat connection, (9.33)-(9.36). The vacuum for SA is designedto undo shift (11.57) and to get back to (11.42)-(11.47), for which it is obvious that anyflat Ω, B = 0 and Aα = Aα = 0 is an exact solution.

Having fixed the vacuum we can look at perturbative expansion

W =W0 +W1 + · · · = Ω +W1 + . . . , (11.82)

B = B0 +B1 + · · · = 0 +B1 + . . . , (11.83)

SA = S0A + S1A + · · · = ZA + S1A + . . . . (11.84)

A general scheme for looking at perturbative series for Vasiliev equations is to first solve forthe algebraic constraints (11.68)-(11.70) and then substitute the solution into space-timepart of equations (11.65)-(11.67). At first order from (11.67) we have

zα ⋆ B1 ⋆ κ +B1 ⋆ κ ⋆ zα = 0 , zα ⋆ B1 ⋆ κ +B1 ⋆ κ ⋆ zα = 0 , (11.85)

that using (11.25) gives

[zα, B1]⋆ = [zα, B1]⋆ = 0 ⇐⇒ ∂B1

∂ZA= 0 . (11.86)

Its generic solution isB1 = C(y, y|x) , (11.87)

where C is an arbitrary z-independent function. We see that at first order B-field appearedto be z-independent by (11.69) — being a manifestation of generic property stating thatz-dependence of B-field is always perturbatively fixed by (11.69). Now the space-timeevolution of C(Y |x) is governed by (11.66) with W = Ω resulting in twisted-adjointflatness condition (9.72). This way we find that at free level master field B indeed describesHS curvatures via (9.72), i.e. DΩC = 0. The next step is to reproduce the on-mass-shelltheorem (9.70) from the linearized equations. In order to do so, we first evaluate S1A from(11.68) which in our case gives

[S0α, Sα1 ]⋆ = −2iC ⋆ κ (11.88)

and similarly for Sα. Substituting S0α from (11.81) and using (11.37), we get

∂Sα1

∂zα= C ⋆ κ = C(−z, y)eizβyβ . (11.89)

98

Before we proceed let us note that there are two types of equations that steadily appearin perturbation analysis, see Appendix 12.7. These are

∂fα

∂zα= g(z) , (11.90)

∂f

∂zα= gα . (11.91)

There is a consistency constraint for the left hand side of (11.91) which requires ∂gα

∂zα= 0.

Generic solutions to (11.90) and (11.91) can be written down in the form of homotopyintegrals

fα =∂η

∂zα+ zα

∫ 1

0

dt t g(tz) , (11.92)

f = c + zα∫ 1

0

dt gα(tz) , (11.93)

where c is z-independent and η is an arbitrary function. The generic solution to (11.89)can be explicitly written now

S1α = ∂αξ1 + zα

∫ 1

0dt t C(−tz, y)eitzβyβ ,

S1α = ∂αξ1 + zα∫ 1

0dt t C(y,−tz)eitzβ yβ ,

(11.94)

where ξ1(Y, Z|x) is arbitrary function that plays a role of gauge ambiguity and can be setto zero for convenience. We prefer, however, to keep them nonzero so as to make surethat they will not affect on-mass-shell theorem (9.70). One can see at this level, that Sindeed is auxiliary being a functional of B. The next step is to substitute (11.94) into(11.67) that gives the following equation

DΩS1A + [W1, S0A] = 0 , (11.95)

where DΩ = d+ [Ω, •]⋆, which is nilpotent DΩ2 = 0 on account of (11.79) and its explicit

action can be easily found from (11.28) to be

DΩf = df + ΩAB(YA − i

∂

∂ZA

) ∂

∂Y Bf . (11.96)

Substituting (11.81) into (11.95) we have

∂αW1 =i

2DΩS1α , ∂αW1 =

i

2D0S1α . (11.97)

These are the equations of type (11.91) and we can solve them as

W1 =i

2DΩξ1 +

i

2zα∫ 1

0

dtDΩS1α|z→tz + c.c.+ ω(y, y|x) , (11.98)

where ω(y, y|x) plays a role of a constant with respect to z in (11.93), while S1 means thatwe explicitly extracted pure gauge dependence from (11.94) which is given by the first

99

term in (11.98). Note that S1α is proportional to zα and, since zαzα ≡ 0, the only termsthat survive are those for which DΩ hits zα inside S1α. There are several such terms,

DΩ ∋ −iωαβ ∂2

∂yα∂zβ− iωαβ ∂2

∂yα∂zβ− ihαα ∂2

∂yα∂zα− ihαα ∂2

∂zα∂yα, (11.99)

which altogether yield

W1 =i

2DΩξ1 + ω(y, y|x)+

+i

2

∫ 1

0

(1− t) dt((

zαωααzαt+ izαh

αα ∂

∂yα

)C(−zt, y)

)eitzβy

β

+ c.c. (11.100)

In accordance with our expectation that physical fields are the initial conditions at Z = 0we find that W1(Y, 0) gives gauge connections ω(Y ) up to a gauge transformation

W1

∣∣∣Z=0

=i

2DΩξ1

∣∣∣∣Z=0

+ ω(y, y|x) . (11.101)

Final step is to substitute (11.100) into (11.65) to determine space-time evolution ofω(y, y|x). At first order this substitution gives for DΩW1 = 0

DΩω(y, y|x) = −DΩ

( i2

∫ 1

0

dtDΩS1α|z→tz + c.c.). (11.102)

Note, that gauge ξ-terms do not contribute to (11.102) as they enter the right hand sidevia DΩ

2ξ ≡ 0. So, indeed, we see that arbitrary function that arises in S field as anintegration constant corresponds to a gauge freedom. Now, the left hand side of (11.102)is explicitly z-independent and so should be its right hand side. In other words, the r.h.s.z-dependence is fictitious as becomes obvious after integration by parts. Equivalently,

∂

∂zαDΩW1 = DΩ

∂

∂zαW1 = DΩ

(i

2DΩS1α

)=i

2DΩDΩS1α = 0 . (11.103)

It makes it convenient to calculate (11.102) at z = 0 and be sure that the result iscorrect. For that reason we only need 2nd derivative terms in (11.96), i.e. (11.99),because otherwise there will be the terms on the r.h.s. of (11.102) proportional to z or zzprior to homotopy integration that do not contribute at z = 0 anyway. Up to irrelevantafter substitution in (11.102) terms we find

W1 = ω(Y )− 1

2hααzα

∂

∂yα

∫ 1

0

dt (1− t)C(−tz, y)hitzβyβ + c.c.+O(z2) (11.104)

Now we need to apply DΩ to (11.104) and set z = 0. Again, the nontrivial part is reachedonly for the second derivative term in (11.96)

(11.102) =− hββ ∧ hαα i2

∂

∂zβzα

∂

∂yβ∂yα

∫ 1

0

dt (1− t)C(0, y) + c.c. = (11.105)

=− i

4hα

α ∧ hαβ ∂2C(0, y)

∂yα∂yβ+ c.c. .

100

So one arrives at the central on-mass-shell theorem (9.70)

DΩω(y, y|x) = −i

4hα

α ∧ hαβ ∂2

∂yα∂yβC(0, y) + c.c. . (11.106)

This completes the free level analysis. It has shown that at linearized approximation theequations do describe bosonic Fronsdal fields along with spin-zero free scalar being a partof HS multiplet in accordance with Sections 9, 10. Moreover, from perturbation theoryit is clear that all degrees of freedom are encoded in a single function C(Y |x). Those asmany as of free fields governed by the Weyl module. It guarantees that at nonlinear levelone has perturbatively the same amount of degrees of freedom, yet the unfolded formof equations (11.65)-(11.70) prevents any field redefinitions that could possibly reducenonlinear equations to the linear ones.

The important question is whether the components of master fields W , B and S canbe eventually treated as space-time tensors in accordance with the equivalence principleor not. Recall that within the unfolded approach the dictionary between fiber fields andworld tensors is achieved by the local Lorentz symmetry. This raises the question whetherthe Lorentz symmetry acts on equations (11.65)-(11.70).

At free level the equation that we got (11.106), contained background Lorentz connec-tions ωαβ and ωαβ via the Lorentz covariant derivative D as a part of the AdS4 covariantderivative DΩ and therefore in a covariant fashion. That makes it possible to convertcomponents of ω(y, y|x) into space time tensors with the aid of frame field. The fact thatequation (11.106) would be Lorentz covariant was not at all obvious from the point of viewof initial equations (11.65)-(11.70) in the first place. Indeed, when fields depend on bothof star-product variables Y and Z, (11.96) no longer acts covariantly. As an example,take f = fαβ(x) y

αzβ , then DΩf = Df − iωαβfαβ which contains Lorentz connection in anoncovariant way. The same problem would be with (11.100) unless noncovariant termsdisappeared from (11.106).

The fact that at free level noncovariant terms for one-forms inW dropped away turnedout to be totally coincidental and is not going to take place any longer already at secondorder. This brings us to the problem of local Lorentz symmetry and laborious search forthe field redefinition that would respect it. Luckily, the Lorentz symmetry happens to beintrinsically resided in the Vasiliev system and this fact is crucially related to the propertyof deformed oscillators (11.63). Moreover, the very existence of Lorentz symmetry to muchextent fixes the equations of motion at the end of the day and could have been a guidelinefor their derivation.

11.5 Manifest Lorentz symmetry

Let us recall the notion of the Lorentz connection. One-form ωαβ is said to be a spin-connection if it transforms under the local Lorentz transformations with parameter Λαβ

as

δωαβ = dΛαβ + ωαγ Λ

γβ + ωβγ Λ

αγ (11.107)

101

Having some generators tαβ of sp(2) we can introduce ω = 12ωαβtαβ , Λ = 1

2Λαβtαβ and

rewrite it as

δω = dΛ + [ω,Λ] (11.108)

An object wα(k) is called a rank-k spin-tensor if it transforms as

δwα(k) = Λαγ w

γα(k−1) . (11.109)

The derivative of wα(k) must be always accompanied by a term with the spin-connectionto give Lorentz-covariant derivative

dwα(k) + ωαγ w

γα(k−1) . (11.110)

We are being somewhat clingy in defining Lorentz connection and local spin-tensors onpurpose as we will face a problem that not every object with indices can be called a spin-tensor. Sometimes it can transform in a wrong way under local Lorentz transformations.For example, instead of (11.110) one may have something like, c.f. (11.100),

ωβγwβγα(k−2) or ωααwα(k) (11.111)

as a contribution to one of the nonlinear equations. These terms can in principle appear.And they do appear. Having such terms would lead to problems in interpretation asthey violate the equivalence principle discussed at the end of the previous section. Thecrucial statement about the Vasiliev equations is that one can find a field redefinition thatremoves (11.111)-like terms and the spin-connection can then be found to appear in theform of Lorentz-covariant derivative only.

To proceed, let us search for the Lorentz generators that rotate master fields properly.At free level we know that these are given by

Lyαβ = − i

2yαyβ , Ly

αβ= − i

2yαyβ . (11.112)

Their action on the free master fields is indeed of Lorentz transformation

Λ =1

2ΛαβLy

αβ + c.c (11.113)

δΛC(y, y|x) = [Λ, C]⋆ =

(Λαβyα

∂

∂yβ+ Λαβ yα

∂

∂yβ

)C(y, y|x) , (11.114)

which when rewritten in component form gives precisely (11.109). The free HS connectionsencoded in ω(y, y|x) transform analogously. Indeed, from

δΛω(y, y|x) = dΛ + [ω,Λ] (11.115)

it follows, that if one decomposes

ω = ω + ωL , ωL =1

2ωLαβL

αβy + c.c. (11.116)

102

in other words if one separates the bilinear in oscillators part of ω from the rest, then onearrives at the Lorentz connection and Lorentz tensor fields transformation

δΛωL = dΛ+ [ωL,Λ] , δΛω = [ω,Λ] . (11.117)

Note also that, while C transforms in the twisted-adjoint under HS transformations(11.72), the Lorentz subalgebra action reduces to the adjoint one, since π does not affectLorentz generators. The transformations (11.112) clearly do not extend to nonlinear levelas they do not act on Z-variable and hence fields that carry indices contracted with ZA

will not be affected by (11.112). This problem can be easily fixed by appending Ly withsimilar generators Lz that rotate z

Lyαβ → L0

αβ = Lyαβ + Lz

αβ = − i2(yαyβ − zαzβ) , (11.118)

[L0αα, f ]⋆ = (yα

∂

∂yα+ zα

∂

∂zα)f , (11.119)

analogously for L. Note, that the sign in (11.118) is important as it guarantees cancelationof second derivative terms. Thus defined generators form Lorentz algebra which properlyrotates spinor oscillators

[L0αβ, L

0γδ]⋆ = ǫβγL

0αδ + . . . , (11.120)

[L0αβ, yγ]⋆ = ǫβγyα + ǫαγyβ , (11.121)

[L0αβ, zγ ]⋆ = ǫβγzα + ǫαγzβ (11.122)

and implies that field B(Y, Z|x) ⋆ κ does transform in Lorentz covariant fashion. Thenonlinear system (11.65)-(11.70) contains field S apart from master fields B and W thaton-shell perturbatively expressed via B-field. We know that in solving for Z-dependenceSα is reconstructed in terms of B ⋆κ up to a gauge freedom. The subtle point is that thegauges chosen in perturbative expansion of Sα should be such that they do not introduceany external objects. This is what makes field S = S[B] purely auxiliary. In other words,at each stage of perturbative expansion with Sα defined up to a ∂αǫ(Y, Z)-term the gaugeparameter ǫ is supposed to contain no extra fields in its z-dependence57. This is obviouslytrue if we impose ∂zαǫ

N = 0 at each stage, so that the exact forms ∂zαǫN(Y, Z) do not

contribute to SNA at any order N . If that is the case S-field is purely auxiliary and should

be properly rotated under Lorentz transformation

δΛSα =δSα

δBδΛB . (11.123)

This is not the case with (11.118). On the contrary one has

[L0αβ , Sγ] = Sαǫβγ + Sβǫαγ +

δSγ

δB[L0

αβ , B]⋆ , (11.124)

57There is a gauge invariance for physical fields which is given by Y -dependent functions only. Thatinvariance is of course unconstrained. The restriction in question concerns the extra gauge freedom inauxiliary Z-sector.

103

where the first two terms on the r.h.s. of (11.124) arise from acting with L0 on the spinorindex of Sα, which can be carried by oscillators and their derivatives. In particular,(11.124) is obvious for S1α, (11.94), provided that ∂zξ1 = 0. This is the point wherethe requirement of no extra fields gets crucial. Therefore, even though (11.118) properlyrotates master field B, it still does not provide one with Lorentz generators for it acts onSα inconsistently with the requirement Sα = Sα[B]. As a result we face the problem ofproper deformation L0 → Lint in such a way as to compensate the extra terms in (11.124)and yet preserve canonical Lorentz action on physical fields. A priori it is not guaranteed,that such a deformation exists and in that case one loses physical interpretation based onequivalence principle. Luckily, for HS system Lorentz symmetry does exist thanks to theproperty of the deformed oscillators (11.63) and (11.64) constructed from S-field. Indeedto compensate unwanted terms we subtract

LSαβ =

i

4Sα, Sβ⋆ (11.125)

from L0αβ and define

Lαβ = L0αβ − LS

αβ (11.126)

Note that in the vacuum Sα = zα we have LS = Lz and hence we are back to L = Ly.Using (11.68)-(11.70) one can check the following commutation relations

[LSαβ , Sγ] = Sαǫβγ + Sβǫαγ , (11.127)

[LSαβ , L

Sγδ] = LS

αδǫβγ + ... [LSαβ , B ⋆ κ] = 0 (11.128)

that give the desired

[Lαβ , Sγ]⋆ =δSγ

δB[L0

αβ , B]⋆ . (11.129)

It is easy to check that L acts on B ⋆ κ in a right way too due to (11.128)

δΛ(B ⋆ κ) ≡[1

2ΛαβLαβ , B ⋆ κ

]

⋆

=

[1

2ΛαβL0

αβ , B ⋆ κ

]

⋆

. (11.130)

Let us make an important comment. The generators (11.126) themselves formally donot form Lorentz algebra in a sense, that [L, L]⋆ 6= L. Indeed, it is straightforward tocheck

[Lαβ, Lγδ]⋆ = (ǫβγLαδ + . . . )−δLS

γδ

δB[L0

αβ , B]⋆ +δLS

αβ

δB[L0

γδ, B]⋆ , (11.131)

where the two last terms break down Lorentz symmetry. This apparent contradiction isin fact false. The reason is the generators (11.126) are field dependent and therefore whensuccessively applied get differ by the corresponding change of the B-field. In other wordsthe action of Lorentz generators (11.126) must account for the change of generators them-selves58. Let us show that being properly treated the successive application of (11.126)

58We are very grateful to M.A. Vasiliev for explaining this point to us.

104

on B-field is equivalent to successive Lorentz transformations. Apply (11.126) to B ⋆ κone time

δΛ1(B ⋆ κ) = [LΛ1 [B ⋆ κ], B ⋆ κ]⋆ = [L0Λ1, B ⋆ κ]⋆ , (11.132)

which as was shown is a proper Lorentz rotation. Now, applying the second time

δΛ2δΛ1(B ⋆ κ) = [LΛ2 [B ⋆ κ], δΛ1(B ⋆ κ)]⋆ + [δΛ1LSΛ2[B ⋆ κ], B ⋆ κ]⋆ , (11.133)

where the second term has been added to account for the change of field dependentgenerator. Now, using that [LS, B⋆κ]⋆ = 0 and therefore [δLS, B⋆κ]⋆+[LS, δ(B⋆κ)]⋆ = 0one finds

(δΛ2δΛ1 − δΛ1δΛ2)(B ⋆ κ) = δ0[Λ2,Λ1](B ⋆ κ) . (11.134)

So, one concludes that local Lorentz symmetry is restored. The situation with one-formsW (Y, Z|x) is a bit trickier. With Λ = 1

2ΛαβLαβ + c.c. looking at

δΛW ≡ dΛ+ [W,Λ]⋆ + c.c. =1

2dΛαβLαβ +

[W,

1

2ΛαβL0

αβ

]

⋆

+ c.c. , (11.135)

where we made use of (11.67), we see that to extract Lorentz tensors and Lorentz con-nection from W we need to decompose it as follows

W = W tensor +WL , WL =1

2ωLαβLαβ +

1

2ωL αβLαβ , (11.136)

such that transformation (11.135) reduces to

δΛWL = dΛ + [WL,Λ]⋆,+c.c. , (11.137)

δΛWtensor =

[W tensor,

1

2ΛαβL0

αβ

]

⋆

+ c.c. , (11.138)

where again we used that in making field dependent Lorentz transformation one has tocompensate the change of generators which effectively cancels the otherwise appearingextra term in (11.137)

ωLγδΛαβδLS

γδ

δ(B ⋆ κ)[L0

αβ , B ⋆ κ]⋆ . (11.139)

Decomposition (11.136) may look like some arbitrary separation of W into two termswhich particularly does not constraint the form of connection fields ωL

αβ and ωLαβ. These

are nonetheless get totally fixed by the requirement that spin-two contribution to be absentinW tensor(Y, Z|x) master field being encoded in the Lorentz connection termWL(Y, Z|x).

In analyzing the problem of Lorentz symmetry we had to impose a requirement of noexternal fields that carry indices to appear in reconstruction of Z-dependence. Does itmean that otherwise the system possesses no local Lorentz symmetry? No it does not,because this extra fields that break down explicit Lorentz symmetry appear in a puregauge fashion. We have proven the existence of local Lorentz symmetry within certainsetup which guarantees (11.124) and allows one to explicitly find Lorentz symmetry. Incase of extra fields there is a gauge transformation that brings solution to the form thatpossesses explicit Lorentz symmetry. It is just that a concrete form of that transformationdepends on the details of particular solution. Or other way around, the form of Lorentzgenerators in that case contains these extra fields in question. One can also have a lookat [38, 110, 113, 114] where the issue of Lorentz symmetry is discussed.

105

11.6 Higher orders

Let us briefly consider how to operate with Vasiliev equations at higher orders. To thesecond order the system acquires the following form

DΩW2 +W 1 ∧ ⋆W 1 = 0 , (11.140)

DΩB2 +W 1 ⋆ B1 −B1 ⋆ W 1 = 0 , (11.141)

DΩS2α + [W 1, S1

α] = −2i∂αW 2 , DΩS2α + [W 1, S1

α] = −2i∂αW 2 (11.142)

2i∂αB2 = S1

α ⋆ B1 +B1 ⋆ S1

α , 2i∂αB2 = S1

α ⋆ B1 +B1 ⋆ ˜S1

α , (11.143)

∂αS2β ǫ

αβ +i

2S1α ⋆ S

1β ǫ

αβ = B2 ⋆ κ , ∂αS2βǫαβ +

i

2S1α ⋆ S

1βǫαβ = B2 ⋆ κ , (11.144)

∂αS2α − ∂αS2

α +i

2[S1

α, S1α]⋆ = 0 , (11.145)

where ∂α = ∂∂zα

and tilde means the twisting, e.g., Sα = π(Sα),˜Sα = π(Sα). Using

(11.143) and (11.93) we can first solve for Z-dependence of B field

2iB2 = zα∫ 1

0

dt(S1α ⋆ C(Y |x) + C(Y |x) ⋆ S1

α)z→tz + c.c.+ C(Y |x) , (11.146)

where S1α is given by (11.94). Substituting (11.146) and (11.98) into (11.141) and after

some algebra and integration by parts we get

DLC − ihαα(yαyα −

∂2

∂yα∂yα

)C + ω ⋆ C − C ⋆ ω+

− 1

2ihααyα

∫ 1

0

dt t(zαC(−tz, y)eitzβy

β

⋆ C − C ⋆ zαC(tz, y)eitzβy

β)+ c.c.

Z=0+

+1

2hαα

∫ 1

0

dt (1− t)(zα

∂

∂yαC(−tt′z, y)eitzβyβ ⋆ C + C ⋆ zα

∂

∂yαC(tz, y)eitzβy

β

+ c.c.)Z=0

= 0 ,

(11.147)

where we also made use of the following formula

∫

[0,1]×[0,1]

dt dt′ t f(tt′) =

∫

[0,1]×[0,1]

f(s)θ(τ − s)dτds =∫ 1

0

(1− s)f(s)ds . (11.148)

Note, that at given order noncovariant terms with respect to Lorentz connection ωαβ thatin principle could have appeared cancel. To proceed to W -sector one solves (11.144),(11.145) for S2

A. To do so, it is convenient to rewrite these equations as

∂AS2B − ∂BS2

A = RAB(B2, S1) , S2

A = ZB

∫ 1

0

dt tRAB(tZ) , (11.149)

which gives

S2α = zα

∫ 1

0

dt t (B2 ⋆ κ − i

4[S1

α, S1β])z→tz + ∂αǫ

2(Y, Z|X)− i

4zα∫ 1

0

dt t [S1α, S

1β]z→tz

106

and allows one to determine Z-dependence of W field from (11.142)

−2iW 2 = zα∫ 1

0

dt (DΩS2α + [W 1, S1

α])z→tz + c.c. + ω2(Y |X) . (11.150)

The latter when substituted into (11.140) results in second order space-time equation forω2(Y |x) that we omit for brevity. Higher-order analysis can be carried out along the samelines. Rewriting the equations to the N -th order one has

DΩWN +

′∑

n+m=N

W n ∧ ⋆Wm = 0 , (11.151)

DΩBN +

′∑

n+m=N

(W n ⋆ Bm − Bm ⋆ W n

)= 0 , (11.152)

DΩSNα +

′∑

n+m=N

[W n, Smα ] = −2i∂αWN , (11.153)

2i∂αBN =

′∑

n+m=N

(Snα ⋆ B

m +Bm ⋆ Snα

), (11.154)

∂αSNβ ǫαβ +

i

2

′∑

n+m=N

Snα ⋆ S

mβ ǫαβ = BN ⋆ κ , (11.155)

∂αSNα − ∂αSN

α +i

2

′∑

n+m=N

[Snα, S

mα ]⋆ = 0 , (11.156)

where some of the equations need to be supplemented with their c.c-versions and theprime

∑′ means that the contribution from the vacuum solution is extracted and writtenseparately. Taking into account our experience in the perturbation theory up to the secondorder, the general recipe consists of circling order by order in the following progression

... −→ (11.154)N −→ (11.155)N −→ (11.153)N −→(11.152)N(11.151)N

−→ (11.154)N+1....

(11.157)which leads to

2iBN = zα∫ 1

0

(′∑

n+m=N

Snα ⋆ B

m +Bm ⋆ Snα)z→tz + c.c.+ CN(Y |X) , (11.158)

SNα = zα

∫ 1

0

dtt(BN ⋆ κ − i

4

′∑

n+m=N

[Snα, S

mβ ] )z→tz + ∂αǫ

N (Y, Z|X)−

− i

4zα∫ 1

0

dtt

′∑

n+m=N

[Snα, S

mβ ]z→tz , (11.159)

WN =i

2zα∫ 1

0

dt(DΩSNα +

′∑

n+m=N

[W n, Smα ])z→tz + c.c. + ωN(Y |X) . (11.160)

107

The solutions to BN and WN are to be used in (11.152)-(11.151) to recover the pertur-bation theory in x-space. The integration constants CN and ωN will play the role of theorder N dynamical fields while other terms will give the interaction vertices.

Let us note that BN and hence CN appears in (11.159) and hence in (11.160) thesame way as in the first order perturbation theory. This results in the same gluing termas in the on-mass-shell theorem but with CN . This is in agreement with the last term in(11.9).

Aknowledgements

We would like to thank Kostya Alkalaev, Nicolas Boulanger, Stefan Fredenhagen, CarloIazeolla, Pan Kessel, Jianwei Mei, Rongxin Miao, Teake Nutma, Per Sundell, MassimoTaronna, Stefan Theisen, all the participants of the Higher Spins, Strings and Dualityschool that took place at Galileo Galilei Institute, Florence. We are especially grateful toMikhail Vasiliev for enlightening discussions, encouragement and comments. We thankGalileo Galilei Institute and organizers of the school for the invitation to give lecturesand for creating a wonderful atmosphere. We thank Kavli Institute for the theoreticalPhysics, Santa-Barbara, for hospitality during the final stage of this work. The work ofE.S. was supported by the Alexander von Humboldt Foundation. The work of V.D. wassupported in part by the grant of the Dynasty Foundation. The work of E.S. and V.D.was supported in part by RFBR grants No. 11-02-00814, 12-02-31837.

12 Extras

12.1 Fronsdal operator on Riemannian manifolds

Let us take the kinetic part of the Fronsdal operator and put it on a general Riemannianmanifold

F [φ]a(s) = φa(s) −∇a∇mφma(s−1) +

1

2∇a∇aφa(s−2)m

m ,

where ∇m is a covariant derivative with respect to some metric gmn. First, we can checkthat the Fronsdal operator still satisfies the double trace constraint,

F [φ]a(s−4)mmnngmmgnn = 0 , (12.1)

which might not have been the case. At least we have the same number of equations asthe number of fields. The Fronsdal operator is not gauge invariant in general, which is asign of a serious problem. Once the gauge symmetry is lost, or weakened, we gained newdegrees of freedom. New degrees of freedom come usually in the form of ghosts. Let usalso note that the Fronsdal operator is not what one gets from the covariantization of theFronsdal action since on has to commute the derivatives in taking the variation.

The source of non-invariance is

∇a[∇a,∇m]ξma(s−2) − [∇a,]ξa(s−1) . (12.2)

108

When the Riemann tensor has only scalar curvature nonvanishing, i.e. we are in (anti)-deSitter, which is displayed by (2.26), the non-invariance is of special form, the same termsoriginate from mass-like terms, as we have already seen. Let us now see what happensif the only non-zero part of the Riemann tensor is the Weyl tensor (which is introducedsystematically in Section 4). Then (12.2) reduces to

2Caaa,mm ξ

a(s−3)mm + Caa,mm∇aξa(s−3)mm − 2Caa,

mm∇mξa(s−2)m , (12.3)

where Caa,bb is the Weyl tensor and Caaa,bb is its first derivative, which is constrained bythe differential Bianchi identity to have -symmetry. We may try to cancel this termsby adding to the Fronsdal operator

Caa,mm φ

a(s−2)mm + gaaCaa,mm φ

a(s−4)mmnn (12.4)

but find this impossible for s > 2, because of the first term (12.3), which does not havederivative on the gauge parameter. The last two terms of (12.3) are problematic toobecause the relative coefficient does not allow one to cancel them by (12.4). So we seethat it is not a piece of cake to make higher-spin fields live on a manifold, which isdifferent from Minkowski or (anti)-de Sitter. In particular generic Ricci flat or Einsteinbackgrounds are not accessible by the Fronsdal theory. The solution, which is a part ofthe Vasiliev theory, is non-minimal in the sense that it cannot be achieved via simplemodifications of the Fronsdal operator.

12.2 MacDowell-Mansouri-Stelle-West

That (anti)-de Sitter algebra (so(d − 1, 2)) so(d, 1) is semi-simple as compared to thePoincare one allows us to improve on the interpretation of gravity as a gauge theoryeven further. Recall that the curvature corresponding to Lab generators of the (anti)-deSitter algebra, (3.40), is Ra,b and Ra,b = F a,b − Λea ∧ eb, where F a,b is the Lab-part ofthe curvature for the Poincare algebra, it is related to the Riemann tensor. It was theobservation by MacDowell and Mansouri, [86] that the following 4d-action

SMM = − 1

2Λ

∫Ra,b ∧ Rc,dǫabcd (12.5)

is equivalent to the Einstein-Hilbert with cosmological term. Indeed, expanding R =F − Λee we find

SMM =

∫ (− 1

2ΛF a,b ∧ F c,d + F a,b ∧ ec ∧ ed − Λ

2ea ∧ eb ∧ ec ∧ ed

)ǫabcd . (12.6)

The first term is of Lovelock type, (3.37), and in fact topological in 4d and hence does notcontribute to the equations of motion. The last two terms sum up to (3.48) with a slightlydifferent cosmological constant, Λ→ Λ/2. Note that Λ→ 0 limit is singular in the actionbut not in the equations of motion since the singularity multiplies the topological term.

The MacDowell-Mansouri action looks similar to the Yang-Mills one, now it is quadraticin the field-strength, but it is not exactly of the Yang-Mills form, (3.39). It can be formally

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rewritten as

SMM =

∫Ra,b ∧ Rc,dǫabcd5 , (12.7)

where we introduced the 5d-epsilon symbol and ǫabcd = ǫabcd5. One may wish to do thatin order to keep the symmetries of the most symmetric solution, so(3, 2) (so(4, 1)), whichis the (anti)-de Sitter space, so it has something to do with rotations in 5d. It does notlook satisfactory yet as one would like to make all symmetries manifest and the presenceof 5 in ǫabcd5 breaks down the symmetry. This can be fixed with a little more work.

First of all, since (anti)-de Sitter algebra (so(d − 1, 2)) so(d, 1) is the algebra of in-finitesimal rotations, it is convenient not to split generators into Lab and Pa, which alreadybreaks the anti-de Sitter symmetry down to the Lorentz one. To accomplish this, anal-ogously to the Lorentz generators Lab themselves, we define TAB = −TBA, where capitalLatin indices A,B, ... run over d+1 values, A = a, 5, where 5 refers to the extra (d+1)-thdirection as compared to the Lorentz algebra. The generators TAB obey

[TAB, TCD] = TADηBC − TBDηAC − TACηBD + TBCηAD , (12.8)

where ηAB is the (so(d−1, 2)) so(d, 1) invariant metric. Lorentz-covariant formulas (3.40)can be recovered upon identifying Lab and Pa with Tab and Ta5. Not surprisingly, if wedefine the Yang-Mills connection Ω = 1

2ΩA,BTAB of the (anti)-de Sitter algebra then the

curvature

RA,B = dΩA,B + ΩA,C ∧ ΩC,B (12.9)

reduces to (3.42) for Ra,b and Ra,5 as well as δΩA,B = DǫA,B reduces to (3.43). In particularthe zero-curvature equation

dΩA,B + ΩA,C ∧ ΩC,B = 0 (12.10)

describes (anti)-de Sitter space analogously to (3.53)-(3.54) provided that we specifiedthe way of splitting ΩA,B into vielbein ea and spin-connection ωa,b, e.g. as Ωa,5 and Ωa,b,and required the vielbein to be nondegenerate. Let us note that (12.10) being a certainflatness condition generates its solution space locally in a pure gauge form. Indeed, onecan easily check that any function g = g(TAB) gives rise to a solution59

Ω = g−1(T )dg(T ) . (12.11)

This fact might suggest an erroneous interpretation, namely that all such solutions aregauge equivalent to Ω = 0 and, particularly, one can “gauge away” the AdS space-timeitself. While formally it seems to be the case, this argument suffers from a flaw thatmakes the vielbein vanish. Recall that only nondegenerate frame fields admit physical

59The expression is somewhat formal. For example, g(T ) can be taken to be the usual exp-map froma Lie algebra to certain neighborhood of the unit element of the group. With the help of g−1g = 1 onechecks that dΩ + ΩΩ = 0, where the product ΩΩ is the usual group product since Ω are given in termsof group element g.

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interpretation. This is another illustration that gauge formalism applied to pure gravityshould be used with great care.

We would like to extend (12.7) to any d and rewrite it in manifestly (anti)-de Sitterinvariant form. A natural extension seems to be of the form, [88],

S =

∫Ra,b ∧Rc,d ∧ ef ∧ ...euǫabcdf...u5 (12.12)

where 5 again means d + 1. However, the Lovelock term∫FFe...e is not topological in

d > 4, which brings in nonlinear corrections to the Einstein equations. It is hard to tell ifthese corrections, however beautiful, are phenomenologically acceptable since we live, atleast effectively, in 4d.

We rewrite (12.12) by using as many uppercase indices as we can

S =

∫RA,B ∧ RC,D ∧ ef ∧ ...euǫABCDf...u5 . (12.13)

Since the value 5 is already occupied A,B,C,D are constrained to the Lorentz index rangesimply because two 5 indices cannot appear simultaneously in the ǫ-symbol. Formally wecan extend ea to the extra direction too, e.g. to define EA such that Ea = ea and E5 = 0since E5 does not contribute to the action anyway. Then we get

S =

∫RA,B ∧RC,D ∧ EF ∧ ...EUǫABCDF...U5 . (12.14)

The anti-de Sitter symmetry is still explicitly broken by 5 in the ǫ-symbol and by em-bedding of ea into EA. A natural way out is to think of 5 as of the vacuum expectationof some compensator vector field, [87], say V A, and we have V A = δA5 . Now it is better,

SMMSW =

∫RA,B ∧RC,D ∧ EF ∧ ...EUV WǫABCDF...UW . (12.15)

We can make all definitions 5-independent by using V A that has to have some vacuumvalue, e.g. δA5 . We can force V A to be non-zero by imposing the following constraint

V AV BηAB = σ , (12.16)

where σ = 1 for de Sitter and σ = −1 for anti-de Sitter. We refer to the choice V A = δA5as the standard gauge.

Introducing a new object V A may seem to be too naive — for some reason we have torestrict ourselves to the range a rather than the full range A, this can be always achievedby introducing a (number of) vectors V A whose purpose is to span the extra directionsof A, so we just have a split Rn+m as Rn ⊕ Rm where (a number of) V A’s span the basisof Rm (in our case n = d, m = 1). Such restoration of ’broken symmetry’ can alwaysbe achieved. In particular instead of extending so(d− 1, 1) to (anti)-de Sitter algebra wecould choose so(d +m) (the signature is immaterial now) with m > 1, which require mlinearly independent V A’s to be introduced. The reason not to go beyond the (anti)-deSitter algebra is our desire to study field theory over (anti)-de Sitter space and we have

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no evidence so far that a larger symmetry that contains so(d− 1, 2) or so(d, 1) is presentin the theory. From this perspective it would be great to have (anti)-de Sitter symmetrymanifest. Actually, we find in Section 10 that a larger symmetry does exists and actson the infinite multiplet of fields of all spins, but not a spin-two alone. But there are nosigns of higher symmetry when dealing with pure gravity60. Indeed, (anti)-de Sitter andMinkowski are known to be maximally-symmetric backgrounds. Therefore, just single V A

should be enough.We need some EA = EA

mdxm such that E5 = 0, Ea = ea and ea = Ωa,5 in the standard

gauge. We can state this as EA = ΩA,BVB but it is better to use

EA = DΩVA = dV A + ΩA,

B VB , (12.17)

because it transforms covariantly. Now all constituents of (12.15) are well-defined. Action(12.15) is manifestly invariant under local (anti)-de Sitter transformations, i.e. Lorentzplus translations, and diffeomorphisms if we assume that the compensator transforms asa fiber vector

δΩA,B = DΩǫA,B , δV A = −ǫA,B V B , δEA = −ǫA,BEB . (12.18)

The local (anti)-de Sitter invariance follows from the fact that all the (anti)-de Sitterindices in (12.15) are contracted using one ηAB or another invariant tensor ǫAB...U of the(anti)-de Sitter algebra and all the constituents transform covariantly. The diffeomor-phism invariance is explicit thanks to differential forms.

Our previous experience shows that having local (anti)-de Sitter transformations anddiffeomorphisms simultaneously is too much. Recall that (3.32) was not invariant underlocal translations unless torsion is zero and at vanishing torsion the local translationsare identical to diffeomorphisms when acting on the frame field ea. Hence we expectd+ d(d− 1)/2 = d(d+ 1)/2 local gauge parameters in total.

The extension of the local symmetry algebra arises due to the presence of the com-pensator field. As we already mentioned, given an action/equations of motion that areinvariant under local transformations belonging to some algebra h and are not invariantunder a bigger algebra g ⊃ h, it is always possible to restore g-symmetry by introduc-ing new fields that transform in such a way as to compensate for g/h-noninvariance. Ofcourse, this does not mean that the theory is invariant under g. A simple example wasgiven above where we could try to extend the local Lorentz symmetry to any so(d+m)with m > 1 by introducing m compensator fields. The genuine symmetry is the stabilityalgebra of the compensator field. The condition for the compensator to remain invariantreads

(Lξ + δǫ)VA = 0 , (12.19)

where we employed both the diffeomorphisms and the local (anti)-de Sitter symmetry. Incomponents it reads

ξm∂mVA − ǫA,B V B = 0 . (12.20)

60In the vicinity of cosmological singularity the Einstein equations were shown to exhibit a highersymmetry, which is a certain infinite-dimensional affine algebra of hyperbolic type, see, e.g, [115]. Thequestion of whether this symmetry acts in the gravity in higher orders and far from the singularity remainsopen.

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It is instructive to look at this condition in the standard gauge V A = δA5 ,

ǫA,B VB = ǫA,5 = 0 . (12.21)

The stability condition for V A kills ǫa,5 components, reducing the local symmetry group todiffeomorphisms and Lorentz rotations with ǫa,b. We are left with the expected amount ofsymmetry, i.e. diffeomorphisms and local Lorentz transformations. Hence, in the standardgauge we roll back to (12.12).

One can choose a nonconstant compensator as well. In the latter case (12.19) startsto mix ξm and local translations ǫA,B V

B. Our physical world (in the tangent space) isto be identified with the subspace orthogonal to the compensator. In particular we haveto ensure that all the quantities, e.g. the vielbein and spin-connection, do not feel thepresence of a compensator. Firstly, note that the generalized vielbein is still effectively ad× d matrix thanks to

EAVA ≡ 0 , (12.22)

where we used V AdVA = 0 and the fact that ΩA,B is antisymmetric. What is the generalizedspin-connection? It can be defined as follows, [88],

ΩLA,B = ΩA,B − 1

V 2(EAV B − EBV A) , D = d+ ΩL . (12.23)

It has some nice properties, which fix its form completely,

DV A ≡ 0 , DEA ≡ 0 , (12.24)

i.e. the compensator V A is insensible with respect to the physical covariant derivative,which is D. The generalized vielbein is covariantly constant, which is an analog of Dea =0. In the standard gauge one recovers Ωa,b

L = Ωa,b.Let us note that pure diffeomorphisms do not preserve EAVA = 0, but we can check

that the combination of diffeomorphisms and gauge symmetries restricted by (12.19) do,

(LξE

A − ǫA,BEB)∣∣

(12.19)= iξR

A,B V

B +D(iξΩA,

B + ǫA,B )VB . (12.25)

The meaning of the compensator field can be understood from the following picture.The (anti)-de Sitter space is replaced with the sphere for simplicity reason. Thinking ofAdS or sphere as d-dimensional hypersurface embedded into (d+1)-dimensional (ambient)space, there is another natural gauge for the compensator, which is to make it lie along theradial direction. The stability algebra of the compensator is then the algebra of rotationsin the plane orthogonal to V A and tangent to the sphere — it is an equivalent of theLorentz algebra.

12.3 Interplay between diffeomorphisms and gauge symmetries

A generic feature of gauge formulation is that diffeomorphisms are entangled with gaugetransformations and hence we cannot treat them separately. When considering gaugefields on manifolds we have to deal with the group of diffeomorphisms61 and group of

61Diffeomorphisms deliver a very difficult infinite-dimensional group to work with.

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Figure 4: Sphere in ambient space and the compensator

V A

x

y

Physical ’Lorentz’ subspace

local gauge transformations. It is easier to talk about the corresponding Lie algebras.The diffeomorphisms diff(M) as a Lie algebra are given by vector fields ξm ∈ diff(M)with the Lie commutator defined to be the Lie bracket of vector fields [ξ, η]m = ξn∂nη

m−ηn∂nξ

m. Gauge parameters, say ǫI(x), are the maps from the given manifold to someLie algebra. Call this linear space of maps h. The bracket is given by the Lie algebracommutator, i.e. [ǫ, θ]I(x) = fI

JKǫJ (x)θK(x).

Gauge parameters are not left intact by diff(M), rather they transform as a numberof scalars under diffeomorphisms, which affect x but do not see the Lie algebra index I.Therefore, the elements of the gauge algebra are affected by diffeomorphisms. Indeed, theLie derivative obeys the Leibnitz law, i.e. Lξ(f(x)g(x)) = (Lξf(x))g(x) + f(x)Lξg(x),where f(x), g(x) are two functions and Lξf(x) = ξm∂mf(x). Then, it is obvious that

Lξ([ǫ, θ]I(x)) = [Lξ(ǫ), θ]

I(x) + [ǫ,Lξ(θ)]I(x) , (12.26)

which means that Lξ acts as a derivation on h. The action of vector fields on gaugeparameters, i.e. the infinitesimal change of coordinates on M , can be understood asthe homomorphism diff(M) → Der(h). Hence pairs (ξm, ǫI) belong to the semidirectproduct diff(M) ⋉ h and the full algebra of symmetries is the semidirect product ofdiffeomorphisms and gauge transformations62.

62There are simpler finite-dimensional examples, the affine group and the Poincare group. The affinegroup Aff(d) consists of pairs (A, a) with A belonging to GL(d) and a being a d-dimensional vector.The group law (A, a) (B, b) = (AB,Ab + a) can be read off the action on vectors (A, a)(~x) = A~x + a.It resembles the semidirect origin of the affine group Aff(d) = GL(d) ⋉ Td, where Td is the group oftranslations in d dimensions, a b = a+ b, a, b ∈ Rd.

114

Given the semidirect structure of the full gauge symmetry algebra it is easy to see thatthe two types of symmetries do not commute with the interesting part of the commutatorresiding in the gauge algebra sector,

[(ξ1, ǫ1), (ξ2, ǫ2)] = ([ξ1, ξ2], [ǫ1, ǫ2] + Lξ1ǫ2 − Lξ2ǫ1) . (12.27)

As we already know, when the Yang-Mills curvature vanishes, every diffeomorphismcan be represented as a gauge transformation. The gauge algebra at a point (just thealgebra we gauge itself) can be larger or smaller than d, which is the dimension of a vectorfield, ξm(x). When the gauge algebra is smaller we cannot identify diffeomorphisms withgauge transformations without having to trivialize the dynamics of gauge fields due tothe necessity of F (A) = 0. For example, this is true for u(1) gauge field, Am. Whenthe gauge algebra is larger one may set only part of the curvature to zero. For example,vanishing of the torsion allows us to treat diffeomorphisms as the subalgebra of localgauge transformations. Note however, that this is true only for the part of the gauge fieldfor which the curvature vanishes, i.e. the vielbein, while the action of a diffeomorphismon the spin-connection is a sum of a gauge transformation and a curvature piece.

Let us consider an example63. The standard gauge transformation law of a spin-onegauge field A ≡ Amdx

m under gauge symmetries with ǫ(x) and diffeomorphisms withξm(x) reads

δξ,ǫA = dǫ+ LξA (12.28)

The commutator of two transformations is a gauge transformation of the same type, whichleads to the following algebra,

[δξ′,ǫ′, δξ,ǫ] = δL[ξ′,ξ],Lξ′ ǫ−Lξǫ′ (12.29)

This form of commutator is typical of the semidirect product.Within field theory we allow for redefinitions of fields and gauge parameters,

ξ → ξ + g(ξ, ∂ξ, ...;φ, ∂φ, ...) , (12.30)

φ→ ξ + f(φ, ∂φ, ...) , (12.31)

where the redefinition of the gauge parameters can be field-dependent and involve deriva-tives of the gauge parameters and fields. Such redefinitions can kill the nice Lie algebrastructure found above, but they are allowed in the realm of field theory, [6]. There is a re-definition that makes the commutator look almost like a direct product. Indeed, insertingǫ− iξA instead of ǫ one finds

δξ,ǫA = dǫ+ iξF (A) , (12.32)

where F (A) = dA. The commutator now reads

[δξ′,ǫ′, δξ,ǫ] = δi[ξ′,ξ],iξiξ′F (A) . (12.33)

Going to the Lie algebras of affine or Poincare group we find [(A, a), (B, b)] = ([A,B], Ab −Ba). Notethat A,B are now any d× d matrices, not necessary invertible. The translations play a role analogous tothat of ’gauge symmetries’ in the sense that they are affected by coordinate transformations associatedwith A. Clearly the vector of translation has to be transformed in accordance with the change of thebasis induced by A.

63We are grateful to T.Nutma and M.Taronna for discussions.

115

The diffeomorphism algebra can be seen in the first argument through the Lie commuta-tor. Miraculously, the gauge parameters ǫ′, ǫ disappeared on the right-hand side. So thecommutator of diffeomorphisms and gauge transformations vanishes now. Unfortunately,we are not able to apply the Lie algebra terminology anymore. Firstly, the ’structureconstants’ became field-dependent via iξiξ′F (A). Secondly, diffeomorphisms do still con-tribute to the sector of gauge transformations via the same iξiξ′F (A).

Therefore, even ignoring the field-dependence of the structure constants we cannot saythat the algebra is a direct product. Though we can say that the deformation of the gaugetransformations induced by coupling of A to gravity is abelian in a sense that there areno ǫ − ξ-mixing terms in the commutator (but we cannot say that it it a direct productin any sense), which is not however the case in (12.29).

Consequently, within the Lie theory the coupling is non-abelian while within the moregeneral framework the coupling is abelian. It is a general situation that the ’structureconstants’ may not be constant and can be field-dependent, so there is no way to interpretit in the language of Lie algebras directly. It is the case for the higher-spin theory. Itturns out that with the help of the unfolded approach, one can assign representationtheory meaning to such cases as well, see Section 6. The unfolded approach disentanglesnontrivial algebraic structure of field theory and diffeomorphisms. A good example ofthe theory where ’structure constants’ are field dependent is provided by an attempt torewrite 3d higher-spin theory in terms of Fronsdal fields, [116].

12.4 Chevalley-Eilenberg cohomology and interactions

Note on cohomologies of Lie algebra. Before going deep into the structure constantswe would like to remind the basic definitions from the theory of Lie algebras. Given a Liealgebra g together with some its representation V we can construct a cochain complexas follows. k-chains are elements of V ⊗ g∗ ∧ ... ∧ g∗, where g∗ is a dual of g as a vectorspace and ∧ denotes the exterior power of g∗ as a vector space. So k-chain Ck is askew-symmetric functional of k elements from g with values in V

Ck : g ∧ ... ∧ g −→ V , (12.34)

c ∈ Ck, c(a1, ..., ak) ∈ V, ai ∈ g , (12.35)

and c(a1, ..., ak) is antisymmetric in its arguments. The differential dk takes Ck to Ck+1

(dkc)(a1, ..., ak+1) =∑

i

(−)iaic(a1, ..., ai, ..., ak+1)+ (12.36)

+∑

i<j

(−)i+j+1c([ai, aj ], a1, ..., ai, ..., aj , ..., ak+1) ,

where aic(...) means that ai acts on the value of c(...) since it belongs to V , which isa g-module. Therefore, given a skew-symmetric functional in k variables from g we canconstruct a skew-symmetric functional in k+1 variables. One can check that dq dq−1 = 0so cohomology groups are defined in a standard way as

Hq = Ker dq/Imdq−1 . (12.37)

116

To compute Hq we have to find a general solution to dqc(a1, ..., aq) = 0 for c ∈ Cq andidentify two solutions if they differ by (dq−1b)(a1, ..., aq) for some b ∈ Cq−1. Classes Hq

are called g-cohomology with values in V .

Cocyles and couplings. Suppose we have an unfolded system with underlying Liealgebra g, which is defined via equations (6.16) for ΩI . We filter the rest of the fieldsby their form degree, i.e. we have sectors of p forms, q forms etc., and without loss ofgenerality we assume that p-forms WA

p take values in some, perhaps trivial, g-moduledefined by the structure constants of one-forms via (6.21), idem for q-forms etc. Considerthe most general unfolded equations that are still linear in WA

p , but can be nonlinear inΩI , i.e. we think of ΩI as the vacuum and consider linear perturbation over it. So theunfolded equations for WA

p have the form of a covariant constancy condition (6.21) whoser.h.s. can be sourced by other fields.

Suppose WAp take values in g-module R1. The simplest option, which we never find

in practice, is to write (p+ 1 = k)

DΩWAp ≡ dWA

p + fIAB Ω

I ∧W Bp = fI1...Ik

AΩI1 ∧ ... ∧ ΩIk . (12.38)

The r.h.s. can be thought of as a skew-symmetric functional fA(Ω, ...,Ω) with values inR1. The integrability (6.4) implies that

fIAB Ω

IfA(Ω, ...,Ω) = kfA(f IJKΩ

J ∧ ΩK , ...,Ω) , (12.39)

i.e. fA(Ω, ...,Ω) is the cocycle of g with values in R1. It is a cocycle but not necessarynontrivial, i.e. not of the form dbk−1 in the notation as above. The trivial cocyles can beremoved by field redefinitions, [92]. Therefore, we see that nontrivial r.h.s. are in one-to-one correspondence with degree-k cohomology of g with values in R1, i.e. are given byHk.

More realistically, suppose we have two sectors WAp and WA

q of p- and q-forms, eachtaking values in some g-modules, say R1 and R2. Assuming p+1 = q+k the most generalequations that are linear in WA

p and WA

q read

DΩWAp ≡ dWA

p + fIAB Ω

I ∧W Bp = fI1...Ik

ABΩI1 ∧ ...ΩIk ∧WB

q , (12.40)

DΩWA

q ≡ dWA

q + fIA

BΩI ∧WB

q = ... , (12.41)

and we ignore the possible r.h.s. in the second equation that will have analogous inter-pretation. The integrability (6.4) implies that fI

AB(Ω, ...,Ω) is a g-cocycle with values in

R1 ⊗R∗2, where R∗

2 is the dual of R2.More generally, nonlinear deformations can look like

DΩWAp ≡ dWA

p + fIAB Ω

I ∧W Bp = fI1...Ik

AB1...Bm

ΩI1 ∧ ...ΩIk ∧WB1q1∧ ... ∧WBm

qm,

where p-form WAp with values in some R is sourced by a number of other formsWB1

q1with

values in R1,..., Rm. The integrability condition implies that fI1...IkAB1...Bm

is a cocyclewith values in R⊗R∗

1 ⊗ ...⊗R∗m.

117

12.5 Universal enveloping realization of HS algebra

Since everything below applies equally well to sp(2n) for any n > 1, while we need sp(4),we shall consider sp(2n) with invariant metric ǫAB. The starting point is the sp(2n)commutation relations (⋆ will denote the product in U(sp(2n)) )

[TAB, TCD]⋆ = TADǫBC + TBDǫAC + TACǫBD + TBCǫAD . (12.42)

Consider the universal enveloping algebra U(sp(2n)) of sp(2n), i.e. the algebra of allpolynomials P (TAB) in the generators TAB modulo the commutation relations (12.42) ofsp(2n). It is easy to work out first several levels by decomposing the Taylor coefficients

P (TAB) = P0 + PAB1 TAB + P

AB|CD2 TAB ⋆ TCD + ... (12.43)

into sp(2n) irreducible tensors. One finds

U(sp(2n)) = •︸︷︷︸0

⊕ ( )︸ ︷︷ ︸1

⊕(

⊕ ⊕ ⊕ •)

︸ ︷︷ ︸2

⊕... (12.44)

The lowest component is the unit 1 of U(sp(2n)), which is an sp(2n) singlet. At the firstlevel we find just TAB. At the second level there is a number of components: the singlet• is the Casimir operator C2 = −1

2TAB ⋆ TAB, it corresponds to P

AB|CD2 = ǫACǫBD +

ǫBCǫAD; is a totally symmetric TAA ⋆ TAA and automatically traceless since ǫAB

is antisymmetric; the antisymmetric component is

=1

2TAB, TC

B ⋆ +1

nǫACC2 . (12.45)

There is also a window component, .In the case of the HS theory in AdS4 we need various totally symmetric sp(4,R)-tensor

of even ranks, which are equivalent to so(3, 2)-tensors with the symmetry of two-rowrectangular Young diagram,

so(5) : s− 1 ⇐⇒ sp(4) : 2s (12.46)

In the HS algebra we are looking for every symmetric sp(4)-tensor of even rank (gener-ators for bosonic fields) must appear once. Already at the second level there are unwantedtensor types. The window component and the rank-two antisymmetric simply do not havethe type we need. The singlet component will lead to proliferation of fields. Indeed, givensome generator T and a singlet C, for any k the monomial T ⋆ Ck is a generator again.

To get rid of the unwanted diagrams one can define a two sided ideal as follows

Iλ = U(sp(2n))⊗(⊕ ⊕ (C2 − λ1)

)⊗ U(sp(2n)) (12.47)

and then try to define the HS algebra g as the quotient

g = U(sp(2n))/Iλ , (12.48)

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with the hope that g is nontrivial and free from unwanted diagrams. Note that whilewe have to get rid of certain nonsinglet components the singlet generator, which is theCasimir, does not necessary have to be put to zero, rather it can be made equivalent tothe unit element. As we see soon, λ turns out to be fixed by the self-consistency of theprocedure.

In practice it means that we try to quotient out or to put to zero certain generatorswithin the universal enveloping algebra. However, U(sp(2n)) is not a free algebra, so atsome point we can come to a contradiction that requiring certain generators to be zeroentail via the commutation relations that TAB ∼ 0 and hence the quotient algebra isempty.

Indeed, let us look at the relations among the low lying generators. First, we combine

the window component and into a single IUU,V V

IUU,V V = TUV ⋆ TUV + γC2

(ǫUU ǫV V +

1

4ǫUV ǫUV

), (12.49)

where the anti-symmetrization over UU and V V is implied wherever necessary. IUU,V V hasthe symmetry of the window Young diagram but it is not completely traceless, containing

as a trace. The coefficient γ = 4/(n(2n+1)) is fixed by requiring IUU,V V not to contain

C2, i.e. ǫUUǫV V IUU,V V = 0.

Using the commutation relations we find the following identities

[TAB, TCB ]⋆ = 2(n+ 1)TAC , (12.50)

TAB ⋆ TCB = αTAC + β ǫACC2 + , α = n+ 1 , β = −1

n, (12.51)

[TAB, TCD]⋆ ⋆ TAB = 4(n+ 1)TCD . (12.52)

Now we would like to study the self-consistency of the procedure. Since IUU,V V belongsto the ideal, any element of U(sp(2n))⊗ IUU,V V must belong too. In particular,

0 ≈ TUU ⋆ IUU,V V = TUV ⋆

(C2

(3

2γ − 2β − 4

)− 2α(α− 2)

). (12.53)

Either we have to fix the Casimir to be C2 − λ1 ∼ 0 with λ = −14n(2n+ 1) or TAB itself

has to be quotient out. The first option leads to a nontrivial solution and fixes λ. If weignore that λ = −1

4n(2n+1) then the resulting two-sided ideal takes away everything, so

the quotient is trivial.It is far from obvious that we will not meet any inconsistencies by considering other

identities in U(sp(2n))/Iλ, but it starts looking as there exist an algebra with exactly thespectrum we need

g = U(sp(2n))/Iλ = • ⊕ ⊕ ⊕ ... (12.54)

More complicated Young shapes have been taken away by , and their descendants

obtained by sandwiching them with U(sp(2n)).

119

We discussed the invariant definition of the HS algebra on the language of the universalenveloping algebra. In particular it is now obvious that HS algebra is an associative onesince it was obtained as a quotient of the associative algebra. However, this realizationis difficult to deal with in practice. There are two sources of complexity. First of all, auniversal enveloping algebra is a complicated object by itself, taking into account it is notfree. Secondly, we quotiented it by a two-sided ideal, which entails many more relationsbetween the generators64.

The advantage of the ⋆-product realization discussed in Section 10 is that the ideal isautomatically resolved. We already noticed that only symmetric tensors appear as Taylorcoefficients in YA, which implies more complicated Young shapes, including the windowand the rank-two antisymmetric, be projected out.

12.6 Advanced ⋆-products: Cayley transform

There are useful tricks that sometimes make star-product calculations quite simple, [57,121]. Particularly, the repetitive gaussian integration drastically simplifies with the aidof the so called Cayley transform. Suppose one wants to compute a bunch of gaussianintegrals of the form

Φ = Φ(f1, ξ1, q1) ⋆ · · · ⋆ Φ(fn, ξn, qn) , (12.55)

where

Φ(f, ξ, q) = exp i(12fABY

AY B + ξAYA + q), (12.56)

Star-product of two such Φ’s is already complicated and given explicitly by

Φ(f1, ξ1, 0) ⋆ Φ(f2, ξ2, 0) =1√

det |1 + f1f2|Φ(f1,2, ξ1,2, q1,2) , (12.57)

where

f1,2 =1

1 + f2f1(f2 + 1) +

1

1 + f1f2(f1 − 1) , (12.58)

ξA1,2 = ξB1

( 1

1 + f2f1(f2 + 1)

)

B

A + ξB2

( 1

1 + f1f2(1− f1)

)

B

A , (12.59)

q1,2 =1

2

( 1

1 + f2f1f2

)

ABξA1 ξ

B1 +

1

2

( 1

1 + f1f2f1

)

ABξA2 ξ

B2 −

( 1

1 + f2f1

)

ABξA1 ξ

B2 . (12.60)

Further multiplication with some other Φ gives cumbersome result. A systematic way toproceed is to use the map of elements (12.55) into SpH(2n) group which is the semidirectproduct of Sp(2n) and Heisenberg group, [122, 123]. Its elements consist of triplets G =(UA

B, xA, c), where UAB ∈ Sp(2n) with the following product

G1 G2 =((U1U2)A

B, x1A + U1ABx2B, c1 + c2 + xA1 U1A

Bx2B

). (12.61)

64Other works along the same lines include [30, 117–120].

120

The embedding into the star-product algebra which respects the SpH(2n) group law

r(G1) Φ(f(G1), ξ(G1), q(G1)

)⋆ r(G2) Φ

(f(G2), ξ(G2), q(G2)

)= (12.62)

= r(G1G2) Φ(f(G1G2), ξ(G1G2), q(G1G2)

)(12.63)

can be shown to be of the following simple form

fAB(G) =(U − 1

U + 1

)

AB, (12.64)

r(G) = 2n/2√det |1 + U |

, (12.65)

ξA(G) = ±2( 1

1 + U

)

A

BxB , (12.66)

q(G) = c+1

2

(U − 1

U + 1

)

ABxAxB . (12.67)

This Cayley transform65 allows one to express (12.55) as

Φ =r(G1...Gn)

r(G1)...r(Gn)Φ(f(G1...Gn), ξ(G1...Gn), q(G1...Gn)

). (12.68)

Let us note, that Cayley transform is not always well defined. Particularly, it is notdefined if for some fAB, fA

CfCB = δA

B. In that case the corresponding Sp(2n) groupelement U does not exist (formally it is at the infinity). Nevertheless, the star-productwith such elements is perfectly well defined as can be seen from (12.58)-(12.60). Suchelements with f 2 = 1 turns out to be star-product projectors

2nei2fABY AY B ∗ 2ne i

2fABY AY B

= 2nei2fABY AY B

, f 2 = 1 . (12.69)

These play an important role in construction of boundary-to-bulk propagators for HSfields and in HS black hole solutions, [55–57].

12.7 Poincare Lemma, Homotopy integrals

There is a standard problem how to solve equations of the form

dfk = gk+1 dgk+1 = 0 (12.70)

where fk ≡ fa[k](x) dxa ∧ ... ∧ dxa is a degree-k differential form and gk+1 has degree

(k+1). This is the simplest system of unfolded type, a contractible pair unless gk+1 ≡ 0.The second equation is the integrability condition for the first one, (12.40). If it is nottrue the system is inconsistent.

65There are two well-known equivariant maps from a Lie algebra to the group, the exp-map and theCayley one. The exp-map is in general neither injective nor surjective. In many cases the Cayley map ismore useful since it is a rational map.

121

The general solution reads

fa[k] =

∫ 1

0

tk dt xcgca[k](tx) +

c, k = 0,

dck−1, k > 0,(12.71)

In checking that this is indeed a solution we used that the number operator xc ∂∂xc gives

the same result as t ∂∂t

on functions g(xt), then one gets a total derivative in t. Also, theintegrability condition needs to be used to transform ∂agca[k] (anti-symmetrization overall a’s is implied) into ∂cga[k].

Viewing the system of equations as an unfolded one there is a gauge symmetry, δf = ξk,δgk+1 = dξk and we see that all solutions are pure gauge from the unfolded perspective66.

Two-dimensional case. In Section 11 we will face a particular case of the above equa-tions, which are the equations along the auxiliary z-direction. That zα is two-dimensionalleaves us with two types of equations, for k = 0 and k = 1 (k = 2 does not have any gk+1

on the r.h.s., so the solution is pure gauge dξ). The first one, k = 0 reads as

∂αf(z, y) = gα(z, y) , (12.72)

where ∂α = ∂∂zα

and y denotes collectively all other variables functions can depend on. Itis implied that gα(z, y) must obey the integrability condition ∂αgα(z, y) = 0. The generalsolution can be represented as

f(z, y) = zα∫ 1

0

dt gα(zt, y) + c(y) (12.73)

The second one we meet, k = 1 reads as

∂αfβ(z, y)ǫαβ = g(z, y) , (12.74)

where g(z, y) is a two form, 12dzα∧dzα g(z, y) and the integrability imposes no restrictions

on g(z, y). The general solution reads

fα(z, y) = zα

∫ 1

0

t dt g(zt, y) + ∂αc(z, y) (12.75)

66It is a choice as to whether think of the equations as having gauge symmetries or not. For example,one can reconstruct Maxwell gauge potential from the field strength by solving dA = F . Despite the factthat F is treated as a two-form, it does not have its own one-form gauge parameter that is capable ofgauging A away completely. So this system is not of unfolded type, see Section 5.4 for the unfolding of aspin-one field. Unfolded equations always assume the richest gauge symmetry possible.

122

A Indices

indices affiliation

m,n, rworld indices of the base manifold, are mostly implicitthanks to the differential form language

a, b, c, ... fiber vector indices of so(d− 1, 1)

α, β, ..., α, β, ... two-component spinor indices

A,B, ...four-component indices of sp(4)-vectors or moregenerally 2n-component indices of sp(2n)

A,B, ...(anti)-de Sitter algebra (so(d− 1, 2)) so(d, 1) indices,range over d+ 1 values

B Multi-indices and symmetrization

Before going to higher-spin fields we need to introduce certain condensed notation forindices. The higher the spin the more indices needed, so it can become a waste of letters.In many cases tensor expressions are symmetric in all the indices, e.g. T abc...u = T bac...u =T acb...u = ..., so it is natural to write T a1a2...as instead of T abc... assuming that the tensoris totally symmetric in all indices ’a’. Moreover, since all indices are denoted by thesame letter now, there is no need to keep them all, to indicate the number thereof issufficient. So a(s) denotes a group of s indices a1a2...as such that an object (tensor) istotally symmetric with respect to all permutations of a1a2...as.

We still need to improve notation a little bit. Sometimes a tensor we find is not totallysymmetric, e.g. ∂mξa(s−1) is only partly symmetric, and we need to make it symmetricby summing over all permutations. Ideologically the right way to achieve this is to apply

the symmetrizator P =1

s!

∑

all permutations

, which has a nice property of being a projector

PP = P . However, in practice it is more useful to adopt a convention where the sumis taken over all necessary permutations without dividing by s!. To simplify notation, allthe indices of the group of indices to be symmetrized are denoted by the same letter, soa string of indices a(k)...a..a means that either the tensor is already symmetric in all ofthem or needs to be symmetrized. For example, (the hatted indices are omitted)

∂aξa(s−1) =i=s∑

i=1

∂aiξa1...ai...as , (B.1)

∂a∂aφa(s−2)mm =

∑

i<j

∂ai∂ajφa1...ai...aj ...asmm . (B.2)

One should be careful with using this convention, still it leads to simpler formulas andmost of the normalization coefficients simply do not appear.

123

For example, let us check the gauge invariance of the Fronsdal operator, (2.4),

δφa(s) = ∂aξa(s−1) , (B.3)

∂mδφma(s−1) = ξa(s−1) + ∂a∂mξ

a(s−2)m , (B.4)

δφa(s−2)mm = 2∂mξ

a(s−1)m , (B.5)

∂a(∂a∂mξa(s−1)m) = 2∂a∂a∂mξ

a(s−1)m . (B.6)

Combining all terms together we find they cancel each other. Notice the bracket removalin the last line brings a factor of two, this is an example of a nested symmetrization.The subtle point is that in ∂a(∂a∂mξ

a(s−1)m) one should think of the inner expression as ageneric rank-(s− 1) symmetric tensor, (despite the fact that it was obtained by summingover s − 1 permutations). It came from ∂a(∂mφ

a(s−1)m) for which s permutations areneeded. Hence, to symmetrize it with ∂a one needs s permutations. In total it givess(s − 1) permutations. However, on the r.h.s. we symmetrize ∂a∂a, which is alreadysymmetric, with ∂mξ

a(s−1)m, so s(s− 1)/2 permutations are needed. This brings a factorof two for the measure of all permutations that have been idle on the l.h.s. becausewe were not aware of the internal structure of (∂a∂mξ

a(s−1)m) and the fact that ∂a∂a isalready symmetric. Nested symmetrizations are the sources of some simple factors thatare important for the final cancelation of terms.

For antisymmetric or to be anti-symmetrized indices we adopt the same rules but withindices enclosed in square brackets, e.g. u[q].

C Solving for spin-connection

Christoffel symbols Γrmn can be solved for as usual. Let us solve for ωa,b. Contracting

(3.15) with embenc and defining Υa|bc = (∂neam − ∂neam)embenc and ωa,b|c = ωa,b

m emc we find

Υa|bc + ωa,c|b − ωa,b|c = 0 . (C.7)

As in the case of Γrnm we add two more equations that differ by cyclic permutation of abc.

Then it is easy to see that

ωa,b|c =1

2

(Υc|ab +Υb|ca −Υa|cb

), (C.8)

and finally

ωa,bm =

1

2

(Υc|ab +Υb|ca −Υa|cb

)ecm . (C.9)

D Differential forms

Among all tensors there is a subclass of covariant totally-antisymmetric tensors thatare special in many respects. Given such a tensor, say Tm1...mq

, Tm1...mimi+1...mq=

−Tm1...mi+1mi...mqfor all i, it is called a differential form and its rank, q, is referred to

as the form degree. Since the index structure is fixed by anti-symmetry it is useful just toindicate the form degree as a subscript, e.g. Tq, without having to write down the indices.

124

A useful way to ensure that the tensor is antisymmetric is to use the Grassmannalgebra. This is an associative algebra with a unit generated by θm obeying θmθn =−θnθm. Consider polynomials in the Grassmann algebra

P (θ) = φ+ Amθm +

1

2Fmnθ

mθn + ...+1

d!ωm1...md

θm1 ...θmd . (D.10)

The expansion coefficients are forced to be antisymmetric tensors and hence the expansionstops at the form of the highest degree possible, which is d. There are three importantoperations on the class of differential forms.

(i) exterior product, which is denoted usually by ∧. This is just the product in theGrassmann algebra. In terms of Taylor coefficients it corresponds to first taking thetensor product and then anti-symmetrizing all the indices. It takes degree-q form Tqand degree-p form Rp to a degree-(p + q) (Tq ∧ Rp) form (note that the order matters)

1(p+q)!

∑σ∈Sn

(−)|σ|σ Tm1...mqRmq+1...mp+q

. In the main text we sometimes omit ∧ symbolwhen one of the factors is a zero-form since zero-forms do not carry any differential formindices and the anti-symmetrization is trivial. Zero-forms serve as purely numerical factorsthat can be commuted without producing a sign, Tq ∧ Rp = (−)pqRp ∧ Tq.

(ii) exterior derivative, d. Operator d is defined as d = θm∂m. It is nilpotent dd ≡ 0and applying θm∇m produces the same result as θm∂m, i.e. Christoffel symbols drop outfrom final expressions because of anti-symmetrization and are irrelevant in definition ofdifferentiation for differential forms.

(iii) inner derivative. Given a vector field ξm one defines the inner derivative iξ asξm ∂

∂θm. The Lie derivative Lξ is then Lξ = diξ + iξd.

In the literature instead of Grassmann algebra, symbols dxm are mostly used whichare required to anti-commute dxm∧dxn = −dxn∧dxm with respect to exterior product ∧.The inner derivative is defined as ξµ ∂

∂dxµ , correspondingly. We will also use dxm havingin mind the Grassmann algebra interpretation.

Integration and Einstein-Hilbert action. The naive way to rewrite∫ √

det g R isto note that67

√det g = det e, and R = Fmn

a,bema enb . There is a fancier way by taking

the advantage of the language of differential forms. Let us recall how to integrate overmanifolds. The measure ddx is not invariant under a change of coordinates, ddx′ =Jddx, where J = det |∂x′/∂x|. One has to cure this non-invariance by multiplying itby something that compensates J . One can take any covariant rank-two tensor, notnecessarily metric and not necessarily symmetric, because its determinant transformsas J−2, so

√det does the job. Another way is to take a rank-d totally-antisymmetric

covariant tensor, say ωm1...md, i.e. a differential form of top degree. As a tensor it has

only one independent component and can be represented as ωm1...md= C(x)ǫµ1...µd

, wherethe totally antisymmetric tensor ǫµ1...µd

is normalized as ǫ12...d = 1. Under change ofcoordinates it transforms as C ′ = det |∂x/∂x′|C = J−1C. Therefore, given a top form,called the volume form sometimes, we can integrate over the manifold. To be precise onemay use ω12...d d

dx as an invariant measure.

67To be honest, det g = det2 e det η.

125

E Young diagrams and tensors

We enter the world of Young diagrams through the GL(d)-tensors’ door and then dis-cuss what needs to be added to cover the case of SO(d) tensors. We do not aim atcomprehensive review and present mostly the facts useful for this particular course.

E.1 Generalities

GL(d). As is well-known, a rank-two tensor T a|b (for simplicity we deal only witheither contravariant or covariant tensors) can be decomposed into its symmetric andantisymmetric components

T a|b = T abS + T a,b

A , (E.11)

T abS = T ba

S =1

2(T a|b + T b|a) , T a,b

A = −T b,aA =

1

2(T a|b − T b|a) . (E.12)

If we deal with GL(d), one can easily see that under any GL(d) transformation

T ab...c −→ Saa′ S

bb′ ...S

cc′ T

a′b′...c′ , Sam ∈ GL(d) , (E.13)

(i) the two parts do not mix and (ii) they preserve their symmetry type, i.e. remain(anti)symmetric. It is also obvious that there are no more GL(d)-invariant conditions onecan impose to decompose T ab

S , T a,bA even further.

It is convenient to employ Young diagrams as a tool to encode graphically the sym-metry type of tensors. A Young diagram is a picture made of boxes, one box per eachtensor index. The rank-two (anti)-symmetric tensors are pictured by the following Youngdiagrams

T abS ∼ a b , T a,b

A ∼ ab . (E.14)

A vector and scalar are denoted by and an empty diagram •, respectively.As for rank-three tensors one finds something new in addition to totally (anti)-

symmetric parts. Since the decomposition of rank-two tensors into irreducibles is known,we can take a rank-three tensor which is already irreducible in some two indices, sayT ab|c = T ba|c. Then one finds,

T ab|c = T abcS +Hab,c

S ,

T abcS =

1

3(T ab|c + T bc|a + T ca|b) , T abc

S = T bacS = T acb

S , (E.15)

Hab,cS =

1

3(2T ab|c − T bc|a − T ca|b) , Hab,c

S = Hba,cS , Hab,c

S +Hbc,aS +Hca,b

S = 0 .

The appearance of the totally symmetric component, T abcS , was expected. There is one

more, Hab,cS , that is neither totally symmetric nor antisymmetric. It is symmetric in

the first two indices, ab and the condition for it not to contain the totally symmetriccomponent, which is already covered by T abc

S , implies that the symmetrization over allthree indices vanishes. With our convention on symmetrization we can rewrite it as

Haa,bS , Haa,a

S = 0 . (E.16)

126

Again one can check that under any GL(d) transformations both T abcS and Hab,c

S preservethe symmetry conditions they obey and do not mix. In the language of Young diagramsthese are pictured as follows

T abcS ∼ a b c Hab,c

S ∼ ca b (E.17)

Because of the shape, Hab,cS is sometimes called ’hook’-diagram. We see tensors tend to

be symmetric in the indices associated with rows of Young diagrams. Not surprisingly, arank-s totally symmetric tensors is denoted by

T a(s) ∼ a a . . . a a︸ ︷︷ ︸s boxes

(E.18)

If instead we take a rank-three tensor that is already antisymmetric in the first twoindices, T ab|c = −T ba|c, we will find a similar decomposition into GL(d) irreducible com-ponents

T ab|c = T abcA +Hab,c

A ,

T abcA =

1

3(T ab|c + T bc|a + T ca|b) , T abc

A = −T bacA = −T acb

A , (E.19)

Hab,cA =

1

3(2T ab|c − T bc|a − T ca|b) , Hab,c

A = −Hba,cA , Hab,c

A +Hbc,aA +Hca,b

A = 0 .

In addition to the totally antisymmetric component T abcA one finds Hab,c

A that is antisym-metric in ab and does not contain the totally antisymmetric component, which is the lastcondition68. In the language of Young diagrams we have

T abcA ∼

cba

Hab,cA ∼ b

a c (E.20)

Tensors tend to be antisymmetric in the indices associated with the columns. A rank-qtotally antisymmetric tensor is denoted by

T u[q] ∼uu

...

uq boxes (E.21)

Tensors that are neither symmetric nor antisymmetric are referred to as having mixed-symmetry. Hab,c

S and Hab,cA have mixed-symmetry. There is a strange thing one might have

noticed that the second diagram in (E.20) is identical to that of (E.17). How come thatHab,c

S and Hab,cA have the same symmetry type but obey different symmetry conditions?

68Coincidentally, for rank-three tensors (E.15) and (E.19) look identical, the symmetry conditions theyobey are different though. This is due to the fact that the cyclic permutation taking place in the definitionof Hab,c

S and Hab,cA (anti)-symmetrizes over the three indices depending on whether the original tensor is

symmetric or antisymmetric in the first two indices. For tensors of higher rank one finds more differencebetween symmetric and antisymmetric presentations.

127

This is an essential feature of genuine mixed-symmetry tensors, i.e. the tensors withthe symmetry of a Young diagram that is different from one-row or one-column diagram.The symmetries of (one-column) one-row Young diagrams are always presented by (anti)-symmetric tensors. There are in general many ways to present a tensor with the symmetryof more complicated Young diagrams. The simplest mixed-symmetry tensor has thesymmetry of . In components we have either some Haa,b

S , which is symmetric in thefirst two indices and obeys the Young condition Haa,a

S = 0, or some Hab,cA , which is

antisymmetric in the first two indices and obeys H[ab,c]A = 0.

Given Hab,cS one can always construct an object of type Hab,c

A as

Hab,cA = Hac,b

S −Hbc,aS . (E.22)

The map is invertible. Indeed, one can go back

Hab,cS = α(Hac,b

A +Hbc,aA ) , (E.23)

where α turns out to be 1/3. Since the map is an isomorphism, it just defines two bases.We can say that there are several ways to implement symmetries of some Young diagraminto a tensor. In practice it is sometimes useful to switch from one base to another one tosimplify computations. It is not possible to have the symmetry properties both of Hab,c

S

and of Hab,cA realized simultaneously.

There are two bases for mixed-symmetry tensors that are most natural, symmetric andantisymmetric. In the (anti)-symmetric base tensors are explicitly (anti)-symmetric inthe indices corresponding to the (columns) rows of Young diagrams, with some additionalrelations imposed. For example, Hab,c

S was given in the symmetric base, while Hab,cA in the

antisymmetric one.Bearing in mind the option of having several ways to represent a tensor, we usually

do not fill the boxes in Young diagrams with indices. For example, there are five differentsymmetry types possible for rank-four tensors

(E.24)

The rows in a Young diagram are left aligned. The length of the rows in a proper Youngdiagram cannot increase downwards (the upper rows are not shorter than the lower ones).

The first and the last diagrams correspond to totally (anti)-symmetric tensors, forwhich there is no ambiguity in the choice of a base. There are two different presentationsfor the three diagrams in the middle. For example, the Riemann and Weyl tensors havethe symmetry of the diagram in the exact center. Usually the Riemann tensor is definedin the antisymmetric base,

RAab,cd = −RA

ba,cd = −RAab,dc = RA

cd,ab , RA[ab,c]d = 0 . (E.25)

Seldom used is the symmetric base

RSab,cd = RS

ba,cd = RSab,dc = RS

cd,ab , RS(ab,c)d = 0 . (E.26)

128

Note that RA(S)ab,cd = R

A(S)cd,ab is not an independent relation and follows from the rest. That

the two ways are equivalent is shown by writing the linear transformation explicitly

RSac,bd = RA

ab,cd +RAcb,ad , RA

ac,bd = β(RSab,cd − RS

cb,ad) , (E.27)

where the first formula is treated as a definition of RS, then the coefficient in the secondformula is found to be β = 1/3. Note that in this particular case it is sufficient to (anti)-symmetrize over two indices, the (anti)-symmetry in the other two indices then followsfrom the properties of the original tensors.

A tensor having the symmetry of the second diagram from (E.24) in the symmetricbase one has T aaa,b that obeys T aaa,a = 0.

Let us consider one more example of a tensor having k symmetry type. We

begin with T a(k−1)|b, which is symmetric in a(k) and there are no symmetry conditionsamong b and a(k). On subtracting the totally symmetric component one finds a remnant,

T a(k)|b = T a(k)b + T a(k),b , (E.28)

T a(k)b =1

k + 1

(T a(k)|b + T a(k−1)b|a

), (E.29)

T a(k),b =1

k + 1

(kT a(k)|b − T a(k−1)b|a

), T a(k),a = 0 . (E.30)

It is not that difficult to get the complete classification of symmetry types. For thepurpose of totally symmetric higher spin fields it is sufficient to restrict ourselves to theclass of two-row Young diagrams with the tensors presented in the symmetric base,

T a(k),b(m) ∼mk =

k︷ ︸︸ ︷a a abb

. . . ab. . .aa

︸ ︷︷ ︸m

(E.31)

The tensor T a(k),b(m) has two groups of indices a(k) and b(m); it is symmetric in k indicesa and m indices b; it is irreducible under GL(d) iff the symmetrization of all indices awith at least one index from the second group vanishes, i.e.

T a(k),ab(m−1) = 0 . (E.32)

Note that one cannot impose more symmetry conditions in general as it would be equiva-lent to requiring a tensor to be symmetric and antisymmetric in some indices at the sametime, which implies the tensor is identically zero. If k < m the tensor vanishes identically,which explains the condition for the length of rows not to increase downwards.

For k = 2, m = 0 we have T ac = T ac, i.e. symmetric. For k = m = 1 we findT a,b + T b,a = 0, i.e. T a,b is antisymmetric. For k = 2, m = 1 we find (E.15). For k = 2,m = 2 we find the symmetry conditions for the Weyl/Riemann tensor, (E.26). There is aslightly degenerate case of rectangular Young diagrams. In the latter case the two groupsof indices in the tensor, T a(k),b(k), are equivalent. In particular, the following relation istrue

T a(k),b(k) = (−)kT b(k),a(k) , (E.33)

129

which is obvious for k = 1 and for k = 2, the Riemann tensor, the condition is also knownto be true. The proof is easy in the antisymmetric presentation of tensors, where one hask pairs of antisymmetric indices and swapping the indices inside all pairs yields (−)k.

At the end of the GL(d) section let us give several identities that are frequently used torearrange indices. Suppose we are given T a(k),b(m) and there is a symmetrization imposedover k indices with one of them taken from the second group of indices, i.e.

T a(k−1)c,ab(m−1) =∑

i

T a1...ai...akc,aib2...bm (E.34)

This is what is effectively imposed when T a(k),b(m) is contracted with another tensor Va(k)that is symmetric

T a(k−1)c,ub(m−1)Va(k−1)u . (E.35)

The defining relation (E.32) then tells that

T a(k−1)c,ab(m−1) = −T a(k),b(m−1)c (E.36)

but

T a(k−1)c,ub(m−1)Va(k−1)u = −1

kT a(k),b(m−1)cVa(k) , (E.37)

the difference in 1kis because (E.34) contains k terms explicitly, while (E.35) does not,

T a(k−1)c,ub(m−1)Va(k−1)u =1

kT a(k−1)c,ab(m−1)Va(k) (E.38)

All k terms in the last expression are identical thanks to the symmetry of Va(k), hencethey cancel 1

k. A more general relation holds true

T a(k−n)c(n),u(n)b(m−n)Va(k−n)u(n) =(−)nn!

k!(k − n)!Ta(k),c(n)b(m−n)Va(k) (E.39)

and allows one to always put all symmetrized indices into the first group of indices. Notethat one can roll symmetrized indices to the group of indices corresponding to a longerrow of Young diagram, but not in the opposite direction. Also note that (E.33) is aparticular case of the identity above for k = n = m.

The property of being a Young diagram tells us that it is the first row/column that isthe longest one. Obviously, the height of the columns in a Young diagram cannot exceedd, for we can choose the antisymmetric base to present a tensor with such a symmetryand it then will carry more than d antisymmetric indices, i.e. it has to vanish.

To draw a line, GL(d) irreducibility requirements impose certain symmetry conditionson the indices carried by a tensor. Irreducibility conditions are nicely encoded by Youngdiagrams. There are in general several ways to present an irreducible GL(d) tensor, onecan transfer between different bases by (anti)-symmetrizing indices.

130

SO(d). In case T a|b is an so(d) tensor (signature is irrelevant) we have an invarianttensor, which is the metric ηab. With the help of the metric one can do more and extractthe trace

T a|b = T abS + T a,b

A +1

dηabT , (E.40)

T abS =

1

2(T a|b + T b|a − 2

dηabT c|dηcd) , T ab

S = T baS , T ab

S ηab = 0 ,

T a,bA =

1

2(T a|b − T b|a) , T a,b

A = −T b,aA ,

T = T c|dηcd .

Again, one can check that the decomposition is stable under any SO(d) transformations.It is obvious that SO(d)-irreducibility requires GL(d)-irreducibility, i.e. Young symme-

try, and one needs to supplement Young symmetry conditions with the trace constraints.The full set of constraints for tensors with the symmetry of two-row Young diagramsincludes

T a(k),b(m) ∼mk , T a(k),ab(m−1) = 0 , (E.41)

T a(k−2)cc,b(m) = 0 , T a(k−1)c,

cb(m−1) = 0 , T a(k),b(m−2)c

c = 0 . (E.42)

There are three types of traces one can take, depending on how the two contracted indicesare distributed over the two groups of indices. Not all of these traces are independent.Indeed, assuming that T a(k−2)c

c,b(m) = 0 we can symmetrize all a(k − 2) with one of the

b’s to see, applying (E.36),

T a(k−2)cc,ab(m−1) = −T a(k−1)c,

cb(m−1) . (E.43)

Symmetrizing now a(k − 1) with one of the b’s once again we find

T a(k−1)c,cab(m−2) = −T a(k),c

cb(m−2) . (E.44)

We see that the first trace condition implies the second and the second then implies thethird, but not in the opposite direction — it is possible to have the third trace conditionssatisfied without enforcing the first and the second.

Young symmetry plus trace constraints furnish the complete set of irreducibility condi-tions in most cases. However, when a tensor has d/2 antisymmetric indices (it is better torefer to the number of rows in the Young diagram), in particular we are in even dimension,one can impose (anti)-selfduality conditions,

T u[q] = ±(i)ǫu[q]v[q] T v[q] , d = 2q , (E.45)

where ǫu[d] is the totally antisymmetric tensor, which is also an invariant tensor of so(d).Whether one can impose the (anti)-self duality condition with ±1 or ±i depends on thedimension, d, modulo 4. In particular one can impose (±i) (anti)-selfduality for tensorswith the symmetry of two-row rectangular Young diagrams in the case of so(3, 1). Thisexplains why a single higher-spin connection ωa(s−1),b(k) splits into two complex conju-gate connections ωα(s+k−1),α(s−k−1), ωα(s−k−1),α(s+k−1) — two-row Young diagrams do not

131

correspond to irreducible tensors in 4d once we allow for a pair of complex conjugatedtensors.

To deal with so(d) is even more restrictive. It is useful to prove the following result(we do not use this anywhere in the lectures): a traceless tensor with the symmetry of aYoung diagram for which the sum of heights of the first two column exceeds d must vanishidentically. Note that nothing prevents the first column to exceed [d/2] at the price of allother columns being shorter than [d/2], for example totally antisymmetric tensors of anyrank q = 0, ..., d do exist.

The existence of ǫu[d] imposes more restrictions on so(d)-Young diagrams. Any tensorhaving a symmetry of a Young diagram whose first column, say of height q, exceeds [d/2]is equivalent to a tensor whose first column does not exceeds [d/2], its height is d− q. Forexample, given a totally antisymmetric tensor T u[q], i.e. its symmetry is given by a Youngdiagram made of a single column, we can dualize it to a rank-(d− q) tensor T ′u[d−q]

T ′u[d−q] = ǫu[d−q]v[q] T

v[q] . (E.46)

This implies that any Young diagram of so(d) should have no more than [d/2] rows.As was just mentioned, this does not mean that all other Young diagrams correspond toidentically vanishing tensors, but those that correspond to nontrivial tensors are equivalentvia ǫu[d]-dualization to Young diagrams with no more than [d/2] rows.

Why Young diagrams? The rationale behind Young diagrams lies in a close connec-tion of representation theory of GL(d) and the symmetric group. The symmetric groupin n letters, Sn, acts naturally on the n-th tensor power, T nV of a vector space, V , bypermuting the factors v1⊗ ...⊗ vn. Hence T nV is a representation of Sn, a reducible one.One can project onto various Sn-irreducible subspaces, which simultaneously projectsonto GL(d) irreducible subspaces. As is well-known the irreducible representations of Sn

are in one-to-one correspondence with conjugacy classes. They can be enumerated bypartitions of n into nonnegative integers, say n = s1 + ... + sn, which can be ordereds1 ≥ s2 ≥ ... ≥ sn ≥ 0. Each partition can be encoded by a Young diagram that hasrows of lengths s1, ..., sn. If we go one further we face the representation theory of Liealgebra and find out that Verma modules are parameterized by a number of constants,the weights, with the number of weights equal to the rank of a given Lie algebra. Inparticular the rank of so(d) is [d/2], so the number of parameters is in accordance withthe number of rows in so(d) Young diagrams. The general theory is discussed in manytextbooks, see e.g. for the summary [124]. The Young diagrams are not just nice picturesand appear naturally in many different topics, [125].

E.2 Tensor products

We more or less understand now what are the conditions for a tensor to be irreducible. Letus discuss the inverse problem of how to decompose a reducible tensor into its irreduciblecomponents. This amount to computing tensor products. The main properties of tensorproducts are associativity, commutativity and distributivity. Actually, this is what wehave already done for the tensors of simplest types. In the case of GL(d) we manually

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found that

(E.12) : T a|b = T abS + T a,b

A ⊗ = ⊕ , (E.47)

(E.15) : T (ab)|c = T abcS +Hab,c

S ⊗ = ⊕ , (E.48)

(E.19) : T [ab]|c = T abcA +Hab,c

A ⊗ = ⊕ . (E.49)

Had we started with the most general rank-three tensor T a|b|c without any symmetryconditions imposed we would have to compute, V ⊗ V ⊗ V ,

⊗ ⊗ (E.50)

We first find that it decomposes according to (E.47) in any pair of indices, say in a andb (tensor product is an associative, commutative operation, so we can insert bracketswherever we like as well as to permute the factors),

( ⊗ )⊗ =(

⊕)⊗ (E.51)

Then, with the help of distributivity and (E.48), (E.49) we get

( ⊗ )⊕(⊗

)= ⊕ 2 ⊕ (E.52)

Using (E.12), (E.15), (E.19) we could obtain a more detailed structure 69. Note that (E.52)does not contain any information about the particular choice of indices that we made andjust states that there is a totally symmetric and a totally antisymmetric component and

two independent components with the symmetry of .

An exercise analogous to (E.15) but for so(d) reads

T aa|c = T aacS +Haa,c

S − 2

(d+ 2)(d− 1)ηaaT b +

d

(d+ 2)(d− 1)ηabT a , (E.53)

T aaaS =

1

3

(T aa|a − 2

d+ 2ηaaT a

), (E.54)

T abcS = T bac

S = T acbS , TS

abb = 0 , (E.55)

Haa,cS =

1

3

(2T aa|c − T ac|a +

1

d− 1(2ηaaT c − ηacT a)

), (E.56)

Hab,cS = Hba,c

S , Hab,cS +Hbc,a

S +Hca,bS = 0 , HS

cc,a = HS

ac,c = 0 .

69There is an analogy with the canonical QM textbook exercise on the su(2) representation theory,namely the multiplication of quantum angular momentum. The decomposition in terms of Young dia-grams contain the same information as the statement

j1 ⊗ j2 = |j1 − j2| ⊕ ...⊕ |j1 + j2| . (1)

But representation denoted loosely by j1 is a (2j1 + 1)-dimensional vector space, analogously for j2 andany of the spins on the r.h.s of (1). In principle we can write the decomposition in more detail, showingexactly how each of the base vectors belonging to one of |j1 − j2 + 2i| on the r.h.s. decomposes into asum of |m1, j1〉⊗ |m2, j2〉. This is analogous to what we did in (E.12), (E.15), (E.19). This more detailedinformation is not captured by (1). Luckily, in many cases knowing (1) or the decomposition in terms ofYoung diagrams is sufficient.

133

Note the appearance of an additional component, the trace T a. We can summarize (E.40),(E.53), and the undone exercise for T [ab]|c with two antisymmetric indices as follows

(E.40) : T a|b ⊗ = ⊕ ⊕ • , (E.57)

(E.53) : T (ab)|c ⊗ = ⊕ ⊕ , (E.58)

left as an exercise : T [ab]|c ⊗ = ⊕ ⊕ . (E.59)

The manipulations with Young diagrams are simpler than writing down the decompositioninto irreducibles in the language of tensors explicitly. For example, for the most generalrank-three tensor T a|b|c we find for V ⊗ V ⊗ V

⊗ ⊗ =(

⊕ ⊕ •)⊗ = ⊕ 2 ⊕ 3 ⊕ (E.60)

As the rank of tensors grows the computation of tensor products become more andmore involved. We will need70

gl(d) : k ⊗ = k ⊕ k + 1 (E.61)

so(d) : k ⊗ = k ⊕ k + 1 ⊕ k − 1 (E.62)

and just for fun

gl(d) :mk ⊗ = m

k⊕

mk + 1 ⊕

m+ 1k (E.63)

so(d) :mk ⊗ = m

k⊕

mk + 1 ⊕

m+ 1k (E.64)

⊕mk − 1 ⊕

m− 1k (E.65)

The general rule for multiplying by is quite simple. For the gl(d) case one tries to addone cell to the diagram in all possible ways such that the result is again a Young diagram.The so(d)-rule is a combination of the gl(d)-rule with an additional cycle where we tryto remove (take the trace) one cell in all admissible ways. One has to remember that ifthe height of some of the Young diagrams on the r.h.s. exceeds [d/2] for so(d) (or d forgl(d)) then one has to use the properties mentioned at the end of gl(d) and so(d) sections.In particular some diagrams may just vanish or should be dualized to fit in [d/2] heightrestrictions in the case of so(d). The general rules, when both factors are generic Youngdiagrams, are quite complicated and can be found, for example, in [126].

E.3 Generating functions

Since in the higher-spin theory we have to work with infinite collections of tensors wefind it convenient, if not necessary, to contract all tensor/spinor indices with some aux-iliary variables. We would like to discuss how various Young/trace constraints can beimplemented on appropriate functional space.

70The tensor product rules do not depend on the signature of the metric, ηab.

134

For example, suppose we need all symmetric tensors, say Ca(k), with tensor of eachrank appearing once, i.e. the space of tensors is multiplicity free. Then all these tensorscan be collected into just one function of auxiliary variables ya

Ca(k) , k = 0, 1, ... ⇐⇒ C(y) =∑

k

1

k!Ca(k)ya...ya . (E.66)

The Taylor coefficients are our original tensors. If our tensors are all traceless then theappropriate functional space is the space of harmonic functions in ya

Ca(k−2)mm = 0 ⇐⇒ C(y) = 0 . (E.67)

Indeed, on computing termwise and equating each Taylor coefficient to zero we get thedesired

C(y) =∂2

∂ym∂ym

∑

k

1

k!Ca(k)ya...ya =

∑

k

1

(k − 2)!Ca(k−2)m

m ya...ya = 0 . (E.68)

Suppose we need a space of tensors with the symmetry of all two-row Young diagrams,again each symmetry type appearing once, i.e.

Ca(k),b(m) Ca(k),ab(m−1) = 0 k = 0, 1, ... m = 0, ..., k (E.69)

Two auxiliary vector-like variables are required now, say ya and pa, with the help of whichwe can build a generating function

C(y, p) =∑

k,m

1

k!m!Ca(k),b(m)ya...ya pb...pb . (E.70)

The Taylor coefficients of a generic function of y and p do not obey any Young symmetryconditions, these are tensors Ca(k)|b(m) symmetric in a(k) and b(m) with no conditionsthat entangle a’s and b’s. With a little thought the right additional restrictions on thefunctional space are found to be

yc∂

∂pcC(y, p) = 0 . (E.71)

Indeed,

yc∂

∂pc

∑

k,m

1

k!m!Ca(k),b(m)ya...ya pb...pb =

∑

k,m

1

k!(m− 1)!Ca(k),cb(m−1)ya...yayc pb...pb (E.72)

all indices contracted with the same commuting variable ya appear automatically sym-metrized, i.e.

Ca(k),cb(m−1)ya...yayc =1

k + 1C(a(k),c)b(m−1)ya...yayc . (E.73)

The original sum was over all values of k and m but the Young symmetry conditionCa(k),ab(m−1) = 0 defines an identically zero tensor for k < m in accordance with therestriction on Young diagrams not to have shorter rows on top of longer ones.

If tensors need to be traceless we can impose trace constraints as harmonicity withrespect to

∂2

∂yc∂yc,

∂2

∂yc∂pc,

∂2

∂pc∂pc. (E.74)

135

F Symplectic differential calculus

Mastering symplectic calculus requires some time and a handful of examples to comparewith. There are formulas that work for any dimension, i.e. with the only assumptionsabout the symplectic metric being

ǫαβ = −ǫβα , ǫαβ = −ǫβα , ǫαβǫγβ = δγα , (F.75)

where δγα us the usual identity matrix. There are also formulas that work in 2d only, i.e.when α, β, ... runs over two values, we shall stress this below.

The main rules are on how to raise and lower indices

yα = ǫαβyβ , yγ = yαǫαγ , (F.76)

with this one checks yα = ǫαβyβ = ǫαβ(yγ ǫγβ) = yα, where we used ǫαβǫγβ = δαγ .If we apply raising/lowering rules to ǫαβ itself we find first of all that ǫαβ is identical

to ǫαβ with all indices raised. Moreover,

ǫαβ = δβα , ǫαβ = −ǫαβ = −δβα . (F.77)

If we remember these rules we can forget about δαβ as an independent object. The minussign in the last expression manifests the antisymmetry of scalar products, e.g. for twovectors vα, uβ, we have

vαuβǫαβ = vαu

α = −vαuβǫβα = −vβuβ . (F.78)

In particular, for any vector vαvα ≡ 0.

Partial derivative ∂α = ∂∂yα

is defined by the following rules

∂αyβ = ǫαβ , ∂αyβ = ǫα

β ≡ δαβ , (F.79)

∂αyβ = ǫαβ , ∂αyβ = ǫαβ = −δαβ , (F.80)

so we can think of anyone as the definition, the rest being consequences of raising/loweringrules. The second one is the most natural. We would like to warn everybody against usingblindly the chain rule to relate ∂α to ∂α. The feature of symplectic calculus is that ∂α isdefined as ∂α with the index raised according to the rules above and it does not coincidewith ∂

∂yα! This is because raising index of ∂α follows a different rule than lowering index

of yα inside ∂/∂yα. It is much more convenient to adopt the same raising/lowering rulesfor all objects than mess up when the rules for variables and derivatives are different. Inparticular, ∂α is required to behave in the same way as any other vector. Another way toovercome the difficulty is to always use ∂α and never try to raise the index. In practicethe chain rule is not needed, the formulae above are sufficient.

The Euler or number operator is useful sometimes

N = yγ∂γ , [N, yα] = yα , [N, ∂α] = −∂α , (F.81)

notice the position of indices since N = −yγ∂γ .

136

In particular, in 2d we have y1 = y2, y2 = −y1 with the canonical choice ǫ12 = 1.

The main property of 2d symplectic world is that any tensor that is antisymmetric in twoindices is proportional to ǫαβ

Tαβ = −Tβα =⇒ Tαβ =1

2ǫαβTγδǫ

γδ =1

2ǫαβT γ

γ (F.82)

Therefore, every two indices can be decomposed as follows

Fαβ =1

2(Fαβ + Fβα) +

1

2(Fαβ − Fβα) = Sαβ +

1

2ǫαβF γ

γ , (F.83)

Sαβ =1

2(Fαβ + Fβα) . (F.84)

The big consequence is that in 2d all nontrivial tensors are symmetric. Whatever anti-symmetric part we find can be expressed in terms of a number of ǫ factors and a totallysymmetric tensor, which is the symplectic trace of the original one.

The last thing is that in higher-spin theory one finds dotted and undotted symplecticindices running over two values, e.g. yα, yα. These are considered as totally independentobjects/indices and they will never mix together (there is no such object as ǫαα) unlessone considers particular solutions to the theory, where the choice of coordinates can breakLorentz symmetry, e.g. (9.37). In fact α and α are different components of the sp(4) indexA, A = α, α. Sometimes we use xαα as a spinorial avatar for xµ, they are related byσααm . The rules for α, α, ... are the same, in particular

∂ααxββ = ǫα

β ǫαβ (F.85)

hence, ∂ααxαα = 4 in accordance with ∂mx

m = 4 in 4d where it applies.

G More on so(3, 2)

Since the anti-de Sitter algebra so(3, 2) is at the core of the 4d higher-spin theory, inparticular the higher-spin algebra can be described in terms of the universal envelopingalgebra U(so(3, 2)), and there is a special isomorphism so(3, 2) ∼ sp(4,R) that simplifiesthings a lot and make the 4d theory special, we would like to give more info on so(3, 2).This section could be too much as at the end we will have representation theory ofso(3, 1) ∼ sl(2,C) and so(3, 2) ∼ sp(4,R) expressed in two different forms each.

G.1 Restriction of so(3, 2) to so(3, 1)

First of all, general arguments from the unfolded approach, see (6.18) and after, tells usthat whenever we find a closed subset of fields of the same form degree, it must formcertain representation of the background space symmetry algebra.

For example this should apply to the set of one-forms ωa(s−1),b(k), k = 0, ..., s − 1.We considered the d-dimensional Minkowski space, but the set does not change when weswitch on the cosmological constant, [95]. So, this set of fields must belong to certain

137

finite-dimensional representations of the anti-de Sitter algebra so(d − 1, 2), which areknown to be tensors or spin-tensors71.

The simplest example of this kind is the pure gravity, where MacDowell-Mansouri-Stelle-West results, see Section 12.2, show that vielbein ea and spin-connection ωa,b canbe viewed as different components of a single so(d, 2)-connection ΩA,B. In this case, thestatement is that a (d + 1) × (d + 1) antisymmetric matrix can be viewed as d × d oneand a d-dimensional vector. In the language of group theory we say that

∣∣∣so(d+1)↓so(d)

∼ ⊕ (G.86)

which is called the branching rules or restriction rules. It is also easy to see that

|so(d+1)↓so(d) ∼ ⊕ ⊕ • (G.87)

Indeed, given a symmetric traceless so(d + 1) tensor TAB we can decompose72 it as thetraceless symmetric tensor T ab− 1

dηabTm

m , vector T a5 and scalar Tmm . Note that so(d+1)

tracelessness implies 0 ≡ TAA = T a

a + η55T55, i.e. T a

a and T 55 are the same up to asign. A less trivial example, which is related to the spin-three case, is

∣∣∣so(d+1)↓so(d)

∼ ⊕ ⊕ (G.88)

On the r.h.s we see exactly the Young symmetries that are needed for the frame-likeformulation of a spin-three field.

The full dictionary up to two-row Young diagrams is as follows

so(d+ 1) tensor so(d) tensor content

• •TA ∼ ⊕ •

TAA ∼ ⊕ ⊕ •

TA(k) ∼ k k ⊕ k − 1 ⊕ ...⊕ ⊕ •

TA,B ∼ ⊕

TA(k),B(k) ∼ k k ⊕ k ⊕ ...⊕ k ⊕ k

TA(m+k),B(m) ∼mm+ k

i=k⊕

i=0

j=m⊕

j=0jm+ i

The most important line is the last but one, which shows that all higher-spin con-nections ωa(s−1),b(k) needed for a spin-s field can be packed into just one connection ofso(d+ 1)

WA(s−1),B(s−1) (G.89)

71Recall that Poincare algebra is not semi-simple and talking about Poincare tensors sounds unnatural.72Again, 5 denotes the extra value of A index that is complementary to the d-dimensional a index,

A = a, 5.

138

which has the symmetry of the two-row rectangular Young diagram of length-(s−1), [88].Again let us note that the branching rules expressed in terms of Young diagrams are

much simpler than the equivalent statements in the language of tensors.

G.2 so(3, 2) ∼ sp(4,R)

The second miraculous isomorphism is between the anti-de Sitter algebra so(3, 2) andsp(4,R). The signature is irrelevant in the section, so one can think of complex Liealgebras, but we do not change the notation. Let us first note that according to thegeneral relation between structure constants of unfolded equations and representationtheory the set of one-forms needed for a spin-s field, i.e.

ωα(m),α(n) m+ n = 2(s− 1) , (G.90)

must belong to some finite-dimensional representation of so(3, 2), see previous section.The same time, as we noticed in Section 8, the same field content can be packed as

ωΩ(2s−2) YΩ...YΩ , (G.91)

where Ω runs73 over four values, which cover α, α. The reason is that so(3, 2) ∼ sp(4,R)and YΩ is a vector of sp(4,R), so both Lorentz algebra so(3, 1) and anti-de Sitter algebraso(3, 2) are special.

To prove the isomorphism and build the dictionary one introduces so(3, 2) Dirac γ-matrices γA ≡ γA

ΛΩ , Λ,Ω, ... = 1, ..., 4, A,B, ... = 0, ..., 4.

(γAγB)ΛΩ + (γBγA)

ΛΩ = 2δΛΩ ηAB (G.92)

The generators of so(3, 2) in the spinorial representation

TAB = −TBA =1

4[γA, γB] (G.93)

can be observed to have a special structure,

TABΛΩ = TAB

ΩΛ , (G.94)

where we raise and lower Λ,Ω, ...-indices with the charge-conjugation matrix CΛΩ = −CΩΛ

using the standard symplectic rules. The charge-conjugation matrix is going to be theinvariant tensor of sp(4,R).

In addition γ-matrices can be shown to be all antisymmetric and CΛΩ-traceless

γAΛΩ = −γAΩΛ , γA

ΛΩCΛΩ = 0 . (G.95)

The latter property implies that one can use γ-matrices to map an so(3, 2)-vector, say VA,into antisymmetric rank-two tensor V ΛΩ = −V ΩΛ, V ΛΩ = γA

ΛΩV A. This is an isomor-phism, which can be proven by observing that V ΛΩ has the same number of components,

73In the main text we use A,B, ... as sp(4,R) indices and A,B, ... as so(3, 2) indices, but this couldcause a confusion when so(3, 2) and sp(4,R) are confronted. Therefore, A,B, ... are changed to Λ,Ω, ....

139

4 · 3/2 − 1 = 5, as V A and one can find a backward transform using the properties ofγ-matrices.

Analogously, a rank-two antisymmetric tensor of so(3, 2), say CA,B = −CB,A can bemapped into rank-two symmetric tensor CΛΩ = CΩΛ, CΛΩ = TΛΩ

ABCA,B. This is an

isomorphism again.Continuing along the same lines, one can derive the following so(3, 2)− sp(4,R) dic-

tionary

so(3, 2) tensor sp(4,R) tensor dim• • 1

Dirac spinor TΛ ∼ 4

TA ∼ TΛ,Ω ∼ 5

TA,B ∼ TΛΛ ∼ 10

TAA ∼ TΛΛ,ΩΩ ∼ 15

TA(k) ∼ k TΛ(k),Ω(k) ∼ k

TA(k),B(k) ∼ k TΛ(2k) ∼ 2k

TA(k),B(m) ∼mk TΛ(k+m),Ω(k−m) ∼

k −mk +m

Irreducible so(3, 2) tensors are traceless with respect to symmetric ηAB and irreduciblesp(4,R) tensors are traceless with respect to antisymmetric CΛΩ.

To summarize, we have the following equivalent ways to describe the space of higher-spin connections of a spin-s field in 4d

so(3, 1) :s−1⊕

k=0

ωa(s−1),b(k) (G.96)

sl(2,C) :⊕

i+j=2(s−1)

ωα(i),α(j) (G.97)

so(3, 2) : ωA(s−1),B(s−1) (G.98)

sp(4,R) : ωΛ(2s−2) (G.99)

(G.96) and (G.98) are valid in any dimension d ≥ 4.

References

[1] C. Fronsdal, Massless fields with integer spin, Phys. Rev. D18 (1978) 3624.

[2] E. S. Fradkin and M. A. Vasiliev, Candidate to the role of higher spin symmetry,Ann. Phys. 177 (1987) 63.

140

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